Core Mathematics - Rwanda Education Board

REPUBLIC OF RWANDA
MINISTRY OF EDUCATION
MATHEMATICS SYLLABUS FOR ADVANCED LEVEL S4-S6
Kigali, 2015
RWANDA EDUCATION BOARD
P.O Box 3817 KIGALI
Telephone : (+250) 255121482
E-mail: [email protected]
Website: www.reb.rw
ADVANCED LEVEL MATHEMATICS SYLLABUS FOR SCIENCES
COMBINATIONS:
- MCB (Mathematics-Chemistry-Biology),
- MCE (Mathematics- Computer Science-Economics)
- MEG (Mathematics-Economics-Geography),
- MPC (Mathematics-Physics-Computer Sciences),
- MPG (Mathematics-Physics- Geography),
- PCM (Physics-Chemistry-Mathematics).
Kigali, 2015
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© 2015 Rwanda Education Board
All rights reserved
This syllabus is the property of Rwanda Education Board, Credit must be provided to the author and source of the document
when the content is quoted.
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FOREWORD
The Rwanda Education Board is honored to avail Syllabuses which serve as official documents and guide to competencebased teaching and learning in order to ensure consistency and coherence in the delivery of quality education across all
levels of general education in Rwandan schools.
The Rwandan education philosophy is to ensure that young people at every level of education achieve their full potential in
terms of relevant knowledge, skills and appropriate attitudes that prepare them to be well integrated in society and exploit
employment opportunities.
In line with efforts to improve the quality of education, the government of Rwanda emphasizes the importance of aligning the
syllabus, teaching and learning and assessment approaches in order to ensure that the system is producing the kind of
citizens the country needs. Many factors influence what children are taught, how well they learn and the COMPETENCES they
acquire, among them the relevance of the syllabus, the quality of teachers’ pedagogical approaches, the assessment strategies
and the instructional materials available. The ambition to develop a knowledge-based society and the growth of regional and
global competition in the jobs market has necessitated the shift to a competence-based syllabus. With the help of the
teachers, whose role is central to the success of the syllabus, learners will gain appropriate skills and be able to apply what
they have learned in real life situations. Hence they will make a difference not only to their own lives but also to the success
of the nation.
I wish to sincerely extend my appreciation to the people who contributed towards the development of this document,
particularly REB and its staff who organized the whole process from its inception. Special appreciation goes to the
development partners who supported the exercise throughout. Any comment of contribution would be welcome for the
improvement of this syllabus.
Mr. GASANA Janvier,
Director General REB.
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ACKNOWLEDGEMENT
I wish to sincerely extend my special appreciation to the people who played a major role in development of this syllabus. It
would not have been successful without the participation of different education stakeholders and financial support from
different donors that I would like to express my deep gratitude.
My thanks first goes to the Rwanda Education leadership who supervised the curriculum review process and Rwanda
Education Board staff who were involved in the conception and syllabus writing. I wish to extend my appreciation to
teachers from pre-primary to university level whose efforts during conception were much valuable.
I owe gratitude to different education partners such as UNICEF, UNFPA, DFID and Access to Finance Rwanda for their
financial and technical support. We also value the contribution of other education partner organisations such as CNLG, AEGIS
trust, Itorero ry’Igihugu, Gender Monitoring Office, National Unit and Reconciliation Commission, RBS, REMA, Handicap
International, Wellspring Foundation, Right To Play, MEDISAR, EDC/L3, EDC/Akazi Kanoze, Save the Children, Faith Based
Organisations, WDA, MINECOFIN and Local and International consultants. Their respective initiative, co- operation and
support were basically responsible for the successful production of this syllabus by Curriculum and Pedagogical Material
Production Department (CPMD).
Dr. Joyce Musabe,
Head of Curriculum and Pedagogical Material Production Department,
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LIST OF PARTICIPANTS WHO WERE INVOLVED IN THE ELABORATION OF THE SYLLABUS
Rwanda Education Board Staff
1. Dr. MUSABE Joyce, Head of Department ,Curriculum and Pedagogical Material Department (CPMD)
2. Mr. RUTAKAMIZE Joseph, Director of Science and Art Unit,
3. Mr. KAYINAMURA Aloys , Mathematics Curriculum Specialist : Team leader,
4. Madame NYIRANDAGIJIMANA Anathalie: Specialist in charge of Pedagogic Norms.
Teachers and Lecturers
1. BIBUTSA Damien, Mathematics teachers at Ecole des Sciences de MUSANZE
2. HABINEZA NSHUTI Jean Clément, Mathematics teachers at Ecole Secondaire de Nyanza
3. NDARA Lula Jerry, Mathematics teachers at EFOTEK
4. UNENCAN MUNGUMIYO Dieudonné, Mathematics teachers at Lycée de Kigali
Other resource persons
Mr Murekeraho Joseph, National consultant
Quality assurer /editors
Dr Alphonse Uworwabayeho (PhD), University of Rwanda (UR), College of Education.
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TABLE OF CONTENTS
FOREWORD ................................................................................................................................................................................................................ 3
ACKNOWLEDGEMENT .............................................................................................................................................................................................. 4
LIST OF PARTICIPANTS WHO WERE INVOLVED IN THE ELABORATION OF THE SYLLABUS ........................................................................... 5
1.
2.
3.
INTRODUCTION: ................................................................................................................................................................................................ 8
1.1.
BACKGROUND TO CURRICULUM REVIEW .............................................................................................................................................. 8
1.2.
RATIONALE OF TEACHING AND LEARNING MATHEMATICS ............................................................................................................... 8
1.2.1.
MATHEMATICS AND SOCIETY.......................................................................................................................................................... 8
1.2.2.
MATHEMATICS AND LEARNERS ...................................................................................................................................................... 9
1.2.3.
COMPETENCES .................................................................................................................................................................................. 9
PEDAGOGICAL APPROACH ............................................................................................................................................................................. 12
2.1.
ROLE OF THE LEARNER .......................................................................................................................................................................... 12
2.2.
ROLE OF THE TEACHER .......................................................................................................................................................................... 14
2.3.
SPECIAL NEEDS EDUCATION AND INCLUSIVE APPROACH ................................................................................................................ 15
ASSESSMENT APPROACH ............................................................................................................................................................................... 15
3.1.
TYPES OF ASSESSMENTS ........................................................................................................................................................................ 16
3.1.1.
FORMATIVE ASSESSMENT: ............................................................................................................................................................ 16
3.1.2.
SUMMATIVE ASSESSMENTS: .......................................................................................................................................................... 16
3.2.
RECORD KEEPING ................................................................................................................................................................................... 17
3.3.
ITEM WRITING IN SUMMATIVE ASSESSMENT ..................................................................................................................................... 17
3.5.
REPORTING TO PARENTS....................................................................................................................................................................... 18
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4.
RESOURCES ...................................................................................................................................................................................................... 18
4.1.
5.
MATERIALS NEEDED FOR IMPLEMENTATION .................................................................................................................................... 18
SYLLABUS UNITS ............................................................................................................................................................................................. 20
5.1.
PRESENTATION OF THE STRUCTURE OF THE SYLLABUS UNITS ...................................................................................................... 20
5.2.
MATHEMATICS PROGRAM FOR SECONDARY FOUR ............................................................................................................................ 21
5.2.1.
KEY COMPETENCES AT THE END OF SECONDARY FOUR ........................................................................................................... 22
5.2.2.
MATHEMATICS UNITS FOR SECONDARY FOUR .......................................................................................................................... 22
5.3.
MATHEMATICS PROGRAM FOR SECONDARY FIVE.............................................................................................................................. 39
5.3.1.
KEY COMPETENCES AT THE END OF SECONDARY FIVE ............................................................................................................. 39
5.3.2.
MATHEMATICS UNITS FOR SECONDARY FIVE ............................................................................................................................ 40
5.4.
MATHEMATICS PROGRAM FOR SECONDARY SIX ................................................................................................................................ 51
5.4.1.
KEY COMPETENCES AT THE END OF SECONDARY SIX ............................................................................................................... 51
5.4.2.
MATHEMATICS UNITS FOR SECONDARY SIX .............................................................................................................................. 52
6.
REFERENCES .................................................................................................................................................................................................... 65
7.
APPENDIX: SUBJECTS AND WEEKLY TIME ALOCATION FOR A’LEVEL .................................................................................................... 66
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1. INTRODUCTION:
1.1. BACKGROUND TO CURRICULUM REVIEW
The motive of reviewing the Mathematics syllabus of advanced level was to ensure that the syllabus is responsive to the
needs of the learner and to shift from objective and knowledge-based learning to competence-based learning. Emphasis in
the review is put more on skills and COMPETENCES and the coherence within the existing content by benchmarking with
syllabi elsewhere with best practices.
The new Mathematics syllabus guides the interaction between the teacher and the learners in the learning processes and
highlights the COMPETENCES a learner should acquire during and at the end of each unit of learning.
Learners will have the opportunity to apply Mathematics in different contexts, and see its important in dailylife. Teachers
help the learners appreciate the relevance and benefits for studying this subject in advanced level.
The new Mathematics syllabus is prepared for all science combinations with Mathematics as core subject where it has to be
taught in seven periods per week.
1.2. RATIONALE OF TEACHING AND LEARNING MATHEMATICS
1.2.1. MATHEMATICS AND SOCIETY
Mathematics plays an important role in society through abstraction and logic, counting, calculation, measurement, systematic
study of shapes and motion. It is also used in natural sciences, engineering, medicine, finance and social sciences. The applied
mathematics like statistics and probability play an important role in game theory, in the national census process, in scientific
research,etc. In addition, some cross-cutting issues such as financial awareness are incorporated into some of the
Mathematics units to improve social and economic welfare of Rwandan society.
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Mathematics is key to the Rwandan education ambition of developing a knowledge-based and technology-led economy
since it provide to learners all required knowledge and skills to be used in different learning areas. Therefore, Mathematics
is an important subject as it supports other subjects.This new curriculum will address gaps in the current Rwanda Education
system which lacks of appropriate skills and attitudes provided by the current education system.
1.2.2. MATHEMATICS AND LEARNERS
Learners need enough basic mathematical COMPETENCES to be effective members of Rwandan society including the ability
to estimate, analyse, interpret statistics, assess probabilities, and read the commonly used mathematical representations and
graphs.
Therefore, Mathematics equips learners with knowledge, skills and attitudes necessary to enable them to succeed in an era of
rapid technological growth and socio-economic development. Mastery of basic Mathematical ideas and calculations makes
learners being confident in problem-solving. It enables the learners to be systematic, creative and self confident in using
mathematical language and techniques to reason; think critically; develop imagination, initiative and flexibility of mind. In
this regard, learning of Matheamtics needs to include practical problem-solving activities with opportunities for students to
plan their own investigations in order to develop their mathematical competence and confidence.
As new technologies have had a dramatic impact on all aspects of life, wherever possible in Mathematics, learners should
gain experience of a range of ICT equipment and applications.
1.2.3. COMPETENCES
Competence is defined as the ability ability to perform a particular task successfully, resulting from having gained an
appropriate combination of knowledge, skills and attitudes.
The Mathematics syllabus gives the opportunity to learners to develop different COMPETENCES, including the generic
COMPETENCES .
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Basic COMPETENCES are addressed in the stated broad subject competences and in objectives highlighted year on year basis
and in each of units of learning. The generic COMPETENCES, basic competences that must be emphasized and reflected in
the learning process are briefly described below and teachers will ensure that learners are exposed to tasks that help the
learners acquire the skills.
GENERIC COMPETENCES AND VALUES
Critical and problem solving skills: Learners use different techniques to solve mathematical problems related to real life
situations.They are engaged in mathematical thinking, they construct, symbolize, apply and generalize mathematical ideas.
The acquisition of such skills will help learners to think imaginatively and broadly to evaluate and find solutions to problems
encountered in all situations.
Creativity and innovation : The acquisition of such skills will help learners to take initiatives and use imagination beyond
knowledge provided to generate new ideas and construct new concepts. Learners improve these skills through Mathematics
contest, Mathematics competitions,...
Research: This will help learners to find answers to questions basing on existing information and concepts and to explain
phenomena basing on findings from information gathered.
Communication in official languages: Learners communicate effectively their findings through explanations, construction
of arguments and drawing relevant conclusions.
Teachers, irrespective of not being teachers of language, will ensure the proper use of the language of instruction by learners
which will help them to communicate clearly and confidently and convey ideas effectively through speaking and writing and
using the correct language structure and relevant vocabulary.
Cooperation, inter personal management and life skills: Learners are engaged in cooperative learning groups to promote
higher achievement than do competitive and individual work.
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This will help them to cooperate with others as a team in whatever task assigned and to practice positive ethical moral values
and respect for the rights, feelings and views of others. Perform practical activities related to environmental conservation
and protection. Advocating for personal, family and community health, hygiene and nutrition and Responding creatively to
the variety of challenges encountered in life.
Lifelong learning: The acquisition of such skills will help learners to update knowledge and skills with minimum external
support and to cope with evolution of knowledge advances for personal fulfillment in areas that need improvement and
development
BROAD MATHEMATICS COMPETENCES
During and at the end of learning process, the learner can:
1. Develop clear, logical,creative and coherent thinking.
2. Master basic mathematical concepts and use them correctly in daily life problem solving;
3. Express clearly, comprehensibly, correctly and precisely in verbal and/orin written form all the reasons and
calculations leading to the required result whenever finding a solution to any given exercise;
4. Master the presented mathematical models and to identify their applications in his/her environment;
5. Arouse learner's mathematical interest and research curiosity in theories and their applications;
6. Use the acquired mathematical concepts and skills to follow easily higher studies (Colleges,Higher Institutions and
Universities);
7. Use acquired mathematical skills to develop work spirit, team work, self-confidence and time management without
supervision;
8. Use ICT tools to explore Mathematics(examples: calculators,computers, mathematical software,…);
9. Demonstrate a sense of research, curiosity and creativity in their areas of study.
MATHEMATICS AND DEVELOPING COMPETENCES
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The national policy documents based on national aspirations identify some ‘basic COMPETENCES’ alongside the ‘Generic
COMPETENCES’’ that will develop higher order thinking skills and help student learn subject content and promote
application of acquired knowledge and skills.
Through observations, constructions, using symbols, applying and generalizing mathematical ideas, and presentation of
information during the learning process, the learner will not only develop deductive and inductive skills but also acquire
cooperation and communication, critical thinking and problem solving skills. This will be realized when learners
make
presentations leading to inferences and conclusions at the end of learning unit. This will be achieved through learner group
work and cooperative learning which in turn will promote interpersonal relations and teamwork.
The acquired knowledge in learning Mathematics should develop a responsible citizen who adapts to scientific reasoning and
attitudes and develops confidence in reasoning independently. The learner should show concern of individual attitudes,
environmental protection and comply with the scientific method of reasoning. The scientific method should be applied with
the necessary rigor, intellectual honesty to promote critical thinking while systematically pursuing the line of thought.
The selection of types of learning activities must focus on what the learners are able to demonstrate such COMPETENCES
throughout and at the end of the learning process.
2. PEDAGOGICAL APPROACH
The change to a competence-based curriculum is about transforming learning, ensuring that learning is deep, enjoyable and
habit-forming.
2.1. ROLE OF THE LEARNER
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In the competence-based syllabus, the learner is the principal actor of his/her education. He/she is not an empty bottle to fill.
Taking into account the initial capacities and abilities of the learner, the syllabus lists under each unit, the activities of the
learner and they all reflect appropriate engagement of the learner in the learning process
The teaching- learning processes will be tailored towards creating a learner friendly environment basing on the capabilities,
needs, experience and interests. Therefore, the following are some of the roles or the expectations from the learners:

Learners construct the knowledge either individually or in groups in an active way. From the learning theory, learners
move in their understanding from concrete through pictorial to abstract. Therefore, the opportunities should be given
to learners to manipulate concrete objects and to use models.

Learners are encouraged to use hand-held calculator. This stimulates mathematics as it is really used, both on job and
in scientific applications. Frequent use of calculators can enhance learners’ understanding and mastering of
arithmetic.

Learners work on one competence at a time in form of concrete units with specific learning objectives broken down
into knowledge, skills and attitude.

Learners will be encouraged to do research and present their findings through group work activities.

A learner is cooperative: learners work in heterogeneous groups to increase tolerance and understanding.

Learners are responsible for their own participation and ensure the effectivness of their work.

Help is sought from within the group and the teacher is asked for help only when the whole group agrees to ask a
question

The learners who learn at a faster pace do not do the task alone and then the others merely sign off on it.
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
Participants ensure the effective contribution of each member, through clear explanation and argumentation to
improve the English literacy and to develop sense of responsibility and to increase the self-confidence, the public
speech ability, etc.
2.2. ROLE OF THE TEACHER
In the competence-based syllabus, the teacher is a facilitator, organiser, advisor, a conflict solver, ...
The specific duties of the teacher in a competence-based approach are the following:

He/she is a facilitator, his/her role is to provide opportunities for learners to meet problems that interest and
challenge them and that, with appropriate effort, they can solve. This requires an elaborated preparation to plan the
activities, the place they will be carried, the required assistance.

He/she is an organizer: his/herrole is to organize the learners in the classroom or outside and engage them through
participatory and interactive methods through the learning processes as individuals, in pairs or in groups. To ensure
that the learning is personalized, active and participative , co-operative theteacher must identify the needs of the
learners, the nature of the learning to be done, and the means to shape learning experiences accordingly

He/she is an advisor: he/she provides counseling and guidance for learners in need. He/she comforts and encourages
learners by valuing their contributions in the class activities.

He/she is a conflict-solver: most of the activities competence-based are performed in groups. The members of a group
may have problems such as attribution of tasks; they should find useful and constructive the intervention of the
teacher as a unifying element.

He/she is ethical and preaches by examples by being impartial, by being a role-model, by caring for individual needs,
especially for slow learners and learners with physical impairments, through a special assistance by providing
remedial activities or reinforncement activities. One should notice that this list is not exhaustive.
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2.3. SPECIAL NEEDS EDUCATION AND INCLUSIVE APPROACH
All Rwandans have the right to access education regardless of their different needs. The underpinnings of this provision
would naturally hold that all citizens benefit from the same menu of educational programs. The possibility of this
assumption is the focus of special needs education. The critical issue is that we have persons/ learners who are totally
different in their ways of living and learning as opposed to the majority. The difference can either be emotional, physical,
sensory and intellectual learning challenged traditionally known as mental retardation.
These learners equally have the right to benefit from the free and compulsory basic education in the nearby
ordinary/mainstream schools. Therefore, the schools’ role is to enrol them and also set strategies to provide relevant
education to them. The teacher therefore is requested to consider each learner’s needs during teaching and learning
process. Assessment strategies and conditions should also be standardised to the needs of these learners. Detailed guidance
for each category of learners with special education needs is provided for in the guidance for teachers.
3. ASSESSMENT APPROACH
Assessment is the process of evaluating the teaching and learning processes through collecting and interpreting evidence of
individual learner’s progress in learning and to make a judgment about a learner’s achievements measured against defined
standards. Assessment is an integral part of the teaching learning processes. In the new competence-based curriculum
assessment must also be competence-based, whereby a learner is given a complex situation related to his/her everyday life
and asked to try to overcome the situation by applying what he/she learned.
Assessment will be organized at the following levels: School-based assessment, District examinations, National assessment
(LARS) and National examinations.
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3.1. TYPES OF ASSESSMENTS
3.1.1. FORMATIVE ASSESSMENT:
Formative assessment helps to check the efficiency of the process of learning. It is done within the teaching/learning process.
Continuous assessment involves formal and informal methods used by schools to check whether learning is taking place.
When a teacher is planning his/her lesson, he/she should establish criteria for performance and behavior changes at the
beginning of a unit. Then at the end of every unit, the teacher should ensure that all the learners have mastered the stated
key unit COMPETENCES basing on the criteria stated, before going to the next unit. The teacher will assess how well each
learner masters both the subject and the generic COMPETENCES described in the syllabus and from this, the teacher will gain
a picture of the all-round progress of the learner. The teacher will use one or a combination of the following: (a) observation
(b) pen and paper (c) oral questioning.
3.1.2. SUMMATIVE ASSESSMENTS:
When assessment is used to record a judgment of a competence or performance of the learner, it serves a summative
purpose. Summative assessment gives a picture of a learner’s competence or progress at any specific moment. The main
purpose of summative assessment is to evaluate whether learning objectives have been achieved and to use the results for
the ranking or grading of learners, for deciding on progression, for selection into the next level of education and for
certification. This assessment should have an integrative aspect whereby a student must be able to show mastery of all
COMPETENCES.
It can be internal school based assessment or external assessment in the form of national examinations. School based
summative assessment should take place once at the end of each term and once at the end of the year. School summative
assessment average scores for each subject will be weighted and included in the final national examinations grade. School
based assessment average grade will contribute a certain percentage as teachers gain more experience and confidence in
assessment techniques and in the third year of the implementation of the new curriculum it will contribute 10% of the final
grade, but will be progressively increased. Districts will be supported to continue their initiative to organize a common test
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per class for all the schools to evaluate the performance and the achievement level of learners in individual schools. External
summative assessment will be done at the end of P6, S3 and S6.
3.2. RECORD KEEPING
This is gathering facts and evidence from assessment instruments and using them to judge the student’s performance by
assigning an indicator against the set criteria or standard. Whatever assessment procedures used shall generate data in the
form of scores which will be carefully be recorded and stored in a portfolio because they will contribute for remedial actions,
for alternative instructional strategy and feed back to the learner and to the parents to check the learning progress and to
advice accordingly or to the final assessment of the students.
This portfolio is a folder (or binder or even a digital collection) containing the student’s work as well as the student’s
evaluation of the strengths and weaknesses of the work. Portfolios reflect not only work produced (such as papers and
assignments), but also it is a record of the activities undertaken over time as part of student learning. . Besides, it will serve
as a verification tool for each learner that he/she attended the whole learning before he/she undergoes the summative
assessment for the subject.
3.3. ITEM WRITING IN SUMMATIVE ASSESSMENT
Before developing a question paper, a plan or specification of what is to be tested or examined must be elaborated to show
the units or topics to be tested on, the number of questions in each level of Bloom’s taxonomy and the marks allocation for
each question. In a competence based curriculum, questions from higher levels of Bloom’s taxonomy should be given more
weight than those from knowledge and comprehension level.
Before developing a question paper, the item writer must ensure that the test or examination questions are tailored towards
competence based assessment by doing the following:





Identify topic areas to be tested on from the subject syllabus.
Outline subject-matter content to be considered as the basis for the test.
Identify learning outcomes to be measured by the test.
Prepare a table of specifications.
Ensure that the verbs used in the formulation of questions do not require memorization or recall answers only but
testing broad COMPETENCES as stated in the syllabus.
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3.4. STRUCTURE AND FORMAT OF THE EXAMINATION
There will be one paper in Mathematics at the end of Primary 6. The paper will be composed by two sections, where the first
section will be composed with short answer items or items with short calculations which include the questions testing for
knowledge and understanding, investigation of patterns, quick calculations and applications of Mathematics in real life
situations. The second section will be composed with long answer items or answers with demonstrations, constructions ,
high level reasonning, analysis, interpratation and drwang conclusions, investigation of patterns and generalization . The
items for the second section will emphasize on the mastering of Mathematics facts, the understanding of Mathematics
concepts and its applications in real life situations. In this section, the assessment will find out not only what skills and facts
have been mastered, but also how well learners understand the process of solving a mathematical problem and whether they
can link the application of what they have learned to the context or to the real life situation. The Time required for the paper
is three hours (3hrs).
The following topic areas have to be assessed: Trigonometry; algebra; analysis; linear algebra; geometry; statistics and
probability.Topic areas with more weight will have more emphasis in the second section where learners should have the
right to choose to answer 3 items out of 5.
3.5. REPORTING TO PARENTS
The wider range of learning in the new curriculum means that it is necessary to think again about how to share learners’
progress with parents. A single mark is not sufficient to convey the different expectations of learning which are in the
learning objectives. The most helpful reporting is to share what students are doing well and where they need to improve.
4. RESOURCES
4.1. MATERIALS NEEDED FOR IMPLEMENTATION
The following list shows the main materials/equipments needed in the learning and teaching process:
-
Materials to encourage group work activities and presentations: Computers (Desk tops&lab tops) and projectors;
Manila papers and markers
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-
Materials for drawing & measuring geometrical figures/shapes and graphs: Geometric instruments, ICT tools such as
geogebra, Microsoft student ENCARTA, ...
-
Materials for enhancing research skills: Textbooks and internet (the list of the textbooks to consult is given in the
reference at the end of the syllabus and those books can be found in printed or digital copies).

Materials to encourage the development of Mathematical models: scientific calculators, Math type, Matlab, etc
The technology used in teaching and learning of Mathematics has to be regarded as tools to enhance the teaching and
learning process and not to replace teachers.
4.2. HUMAN RESOURCE
The effective implementation of this curriculum needs a joint collaboration of educators at all levels. Given the material
requirements, teachers are expected to accomplish their noble role as stated above. On the other hand school head teachers
and directors of studies are required to make a follow-up and assess the teaching and learning of this subject due to their
profiles in the schools. These combined efforts will ensure bright future careers and lives for learners as well as the
contemporary development of the country.
In a special way, the teacher of Mathematics at ordinary level should have a firm understanding of mathematical concepts at
the leavel he / she teaches. He/she should be qualified in Mathematics and have a firm ethical conduct. The teacher should
possess the qualities of a good facilitator, organizer, problem solver, listener and adviser. He/she is required to have basic
skills and competence of guidance and counseling because students may come to him or her for advice.
Skills required for the Teacher of Religious Education
The teacher of Mathematics should have the following skills, values and qualities:
-
Engage learners in variety of learning activities
Use multiple teaching and assessment methods
Adjust instruction to the level of the learners
Have creativity and innovation the teaching and learning process
Be a good communicator and organizer
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-
Be a guide/ facilitator and a counsellor
Manifest passion and impartial love for children in the teaching and learning process
Make useful link of Mathematics with other Subjects and real life situations
Have a good master of the Mathematics Content
Have good classroom management skills
5. SYLLABUS UNITS
5.1. PRESENTATION OF THE STRUCTURE OF THE SYLLABUS UNITS
Mathematics subject is taught and learnt in advanced level of secondary education as a core subject, i.e. in S4, S5 and S6
respectively. At every grade, the syllabus is structured in Topic Areas, sub-topic Areas where applicable and then further
broken down into Units to promote the uniformity, effectivness and efficiency of teaching and learning Mathematics. The
units have the following elements:
1. Unit is aligned with the Number of Lessons.
2. Each Unit has a Key Unit Competence whose achievement is pursued by all teaching and learning activities
undertaken by both the teacher and the learners.
3. Each Unit Key Competence is broken into three types of Learning Objectives as follows:
a. Type I: Learning Objectives relating to Knowledge and Understanding (Type I Learning Objectives are also
known as Lower Order Thinking Skills or LOTS)
b. –Type II and Type III: These Learning Objectives relate to acquisition of skills, Attitudes and Values (Type II
and Type III Learning Objectives are also known as Higher Order Thinking Skills or HOTS) – These Learning
Objectives are actually considered to be the ones targeted by the present reviewed curriculum.
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4. Each Unit has a Content which indicates the scope of coverage of what to be tought and learnt in line with stated
learning objectives
5. Each Unit suggests a non exhaustive list of Learning Activities that are expected to engage learners in an interactive
learning process as much as possible (learner-centered and participatory approach).
6. Finally, each Unit is linked to Other Subjects, its Assessment Criteria and the Materials (or Resources) that are
expected to be used in teaching and learning process.
The Mathematics syllabus for ordinary level has got 7 Topic Areas: Trigonometry, Algebra, Analysis, Linear algebra,
Geometry, Statistics and Probability and these topic areas are found in each of the three grades of the advanced level
which are S4, S5 and S6. As for units, they are 16 in S4, 10 in S5 and 9 in S6
5.2. MATHEMATICS PROGRAM FOR SECONDARY FOUR
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5.2.1. KEY COMPETENCES AT THE END OF SECONDARY FOUR
After completion ofsecondary 4, the mathematics syallabus will help the learner to:
1. Use the trigonometric concepts and formulas in solving problem related to trigonometry;
2. Think critically and analyze daily life situations efficiently using mathematical logic concepts and infer conclusion.
3. Model and solve algebraically or graphically daily life problems using linear, quadratic equations or inequalities.
4. Represent graphically simple numerical functions.
5. Perform operations on linear transformation and solve problems involving geometric transformations.
6. Determine algebraic representations of lines, straight lines and circles in the 2D
7. Extend understanding, analysis and interpretation of data arising from problems and questions in daily life to include
the standard deviation.
8. Use matrices and determinants of order 2 to solve systems of linear equations and to define transformations of 2D
9. Extend understanding, analysis and interpretation of data arising from problems and questions in daily life to include
the standard deviation.
10. Use counting techniques and concepts of probability to determine the probability of possible outcomes of events
occurring under equally likely assumptions
5.2.2. MATHEMATICS UNITS FOR SECONDARY FOUR
22
Topic Area: TRIGONOMETRY
S4- MATHEMATICS
Sub-topic Area: TRIGONOMETRIC CIRCLE AND IDENTITIES
Unit 1 : Fundamentals of trigonometry
No. of lessons:26
Key unitCompetence:Use trigonometric circle and identities to determine trigonometric ratios and apply them to solve related problems
Learning Objectives
Contents
Learning Activities
Attitudes and
values
 Represent
 Appreciate
 Trigonometric
 Mental task – imagine a point on the edge of a wheel – as the
graphically sine,
the
concepts:
wheel turns how high is the point above the centre? – sketch the
cosine and
relationship
graph
 Angle and its
tangent, functions
between the
 Practical – on graph paper draw circle radius 10cm and
measurements
and, together
trigonometric
measure half chord length and distance from centre to chord for
 Unit circle
with the unit
values for
angles (say multiples of 15o) – plot the graphs – use calculator
 Trigonometric
circle, use to
different
to determine which is sine and cosine. What is the radius of the
ratios
relate values of
angles
calculator’s circle? – unit circle
 Trigonometric
any angle to the
 Verify

Use of dynamic geometry and graph plotting to illustrate
identities
value for a
reasonablene
relationship e.g. geogebra
 Reduction to
positive acute
ss of answers
functions of Positive  In groups use unit circle and graphs to determine the
angle
when solving
relationship between trigonometric functions of any angle and
acute angles
 Use
problems
the value for an acute positive angle – generalize.
 Triangles and
trigonometry,
 Group investigation -What angle subtends an arc length equal
Applications:
including the sine
to the radius? – define a radian, make a table of equivalences
 Bearing
and cosine rules,

Derive trigonometric identities, sine and cosine rules
 Air Navigation
to solve problems

Apply trigonometry to practical problems involving triangles
 Inclined plane…
involving
and angles.
triangles
Links to other subjects: Physics (optics, wave, electricity), Geography, Architecture, Engineering,...
Assessment criteria: Apply trigonometric concepts to solve problems involving triangles and angles
Materials:Geometricinstruments (ruler,T-square,compass), graph papers, digital technology including calculators
Knowledge and
understanding
 Define sine,
cosine, and
tangent (cosecant,
secant and
cotangent) of any
angle – know
special values
(30o, 45o, 60o)
 Convert radians to
degree and vice
versa.
 Differentiate
between
complementary
angles,
supplementary
angles and coterminal angles
Skills
Topic Area: ALGEBRA
Sub-topic Area: MATHEMATICAL LOGIC AND APPLICATIONS
23
S4 - MATHEMATICS
Unit 2 :Propositional and predicate logic
No. of lessons: 14
Key unit Competence: Use mathematical logic to organize scientific knowledge and as a tool of reasoning and argumentation in daily life
Learning Objectives
Knowledge and
understanding
 Distinguish between
statement and
proposition
 Convert into logical
formula composite
propositions and vice
versa
 Draw the truth table of a
composite proposition
 Recognize the most
often used tautologies
(E g. De Morgan’s Laws)
Skills
Attitudes and values
 Use mathematical logic to
infer conclusion from
given proposition
 Evaluate claims, issues and
arguments, and identify
mistakes in reasoning and
prove the validity or
invalidity of arguments in
ordinary discourse.
 Show that a given logic
statement is tautology or a
contradiction.
 Judge situations accurately
and act with equality
 Observe situations and
make appropriate decisions
 Appreciate and act with
thoughtfulness: grasp and
demonstrate carefulness…
 Develop and show mutual
respect
 Demonstrate brodmindedness
Content
Learning Activities
 Generalities:
 Introduction and
fundamental definitions
 Propositional logic:
 Truth tables
 Logical connectives
 Tautologies and
Contradictions
 Predicate Logic:
 Propositional functions
 Quantifiers
 Applications:
 Set theory
 Electric circuits …
 Practical - Deduce
if given
statement is or
not a proposition
 Group
investigationMake a research
in advance in the
library about
propositional
logic
Links to other subjects:Electricity (Use of De Morgan’s Laws), Biology,
Assessment criteria: Learner is able to:Use mathematical logic to organize scientific knowledge and as a tool of reasoning and argumentation in daily life
Materials: Manila papers,…
Topic Area: ALGEBRA
Sub-topic Area: NUMBER AND OPERATIONS
24
S4- MATHEMATICS
Unit 3:Binary operations
No. of lessons:14
Key unit Competence:Use mathematical logic to understand and perform operations using the properties of algebraic structures
Learning Objectives
Knowledge and
understanding
 Define a group, a ring , an
integral domain and a field
 Demonstrate that a set ( or
is not) a group, a ring or a
field under given operations
 Demonstrate that a subset
of a group is (or is not ) a
sub group
Skills
Attitudes and values
 Determine the properties of
a given binary operation
 Formulate, using adequate
symbols, a property of a
binary operation and its
negation
 Construct the Cayley table of
order 2,3,4
 Discover a mistake in an
incorrect operation
 Appreciate the
importance and the
use of properties of
binary operations
 Show curiosity,
patience, mutual
respect and
tolerance in the
study of binary
operations
Content
 Definitions and
Properties
 Algebraic structures
Links to other subjects:
- Physics (Electricity in calculation of numerical values from a formula)
- Entrepreneurship (economics:calculation of data from sales), Computer science, …
Assessment criteria:
- Explain why grouping, interchanging, distributing,... is correct or not depending on the context
- Carry out binary operations and determine their properties
Materials:Digital technology including calculators
25
Learning Activities
 Discuss in groups, patiently in
mutual respect and tolerance,
the main facts about binary
operations and algebraic
structures and choose a
reporter to present the work
 With an example of binary
operation define a group, ring,
field
Topic Area: ALGEBRA
Sub-topic Area: NUMBERS AND OPERATIONS
S4 - MATHEMATICS
Unit 4:SET
OF REAL NUMBERS
No. of lessons:24
Key Unit Competence: Think critically using mathematical logic to understand and perform operations on the set of real numbers and its
subsetsusing the properties of algebraic structures
Learning Objectives
Knowledge and
understanding
 Match a number
and the set to
which it belongs
Content
Attitudes and
values
 Appreciate the  Absolute value
importance
and its
and the use of
properties
properties of
 Powers and
operations on
radicals
real numbers
 Decimal
 Show curiosity
logarithms and
for the study of
properties.
operations on
real numbers
Skills
Learning Activities
 Classify numbers into naturals, integers, rational and
 Group investigation –
irrationals
Make research in advance
 Determine the restrictions on the variables in
in the library about Sets of
rational and irrational expressions
numbers (natural numbers,
 Illustrate each property of a power, an exponential ,a
integers, rational numbers
 Define a power, an
radical, a logarithm, the absolute value of a real
and irrational numbers
exponential ,a
number
 Mental task What are the
radical, a
 Use logarithm and exponentials to model simple
main facts about sets R of
logarithm, the
problems about growth, decay, compound interest,
real numbers
absolute value of a
magnitude of an earthquake ...
 Apply operations on set of
real number
 Transform a logarithmic expression to equivalent
real numbers to illustrate
power or radical form and vice versa
relation to arithmetic
 Rewrite an expression containing “absolute value”
using order relation
Links to other subjects:Physics: e.g. converting temperature from degree Celsius to degree Fahrenheit, converting seconds to minutes and vice versa e.g.
Entrepreneurship and in Economics Organisation and computation of data from sales ,Chemistry: e.g. The decay process Biology: e.g. growth of bacteria
,Geography: e.g. magnitude of an earthquake
Assessment criteria: Use mathematical logic to understand and perform operations on the set of real numbers and its subsets using the properties of algebraic
structures
Materials:Graph papers, manila papers, digital technology including calculators,...
26
Topic Area: ALGEBRA
Sub-topic Area: EQUATIONS AND INEQUALITIES
S4 - MATHEMATICS
Unit 5:Linear equations and inequalities
No. of lessons:12
Key unitCompetence: Model and solve algebraically or graphically daily life problems using linear equations or inequalities
Learning Objectives
Knowledge and
understanding
 List and clarify
the steps in
modeling a
problem by
linear equations
and inequalities
Skills
Attitudes and values
 Equations and inequalities in one
unknown
 Parametric equations and
inequalities in one unknown.
 Simultaneous equations in two
unknowns ...
 Applications:
 Economics ( Problems about
supply and demand analysis, … )
 Physics ( Linear motions, Electric
circuits, …)
 Chemistry ( Balancing
equations...)
 Appreciate, value and
care for situations
involving to quadratic
equations and
quadratic inequalities
in daily life situation
 Show curiosity about
quadratic equations
and quadratic
inequalities
Content
Learning Activities
 Equations and inequalities in one
unknown
 Parametric equations and
inequalities in one unknown.
 Simultaneous equations in two
unknowns ...
 Applications:
 Economics ( Problems about
supply and demand
analysis,…)
 Physics ( Linear motions,
Electric circuits, …)
 Chemistry ( Balancing
equations, …)


Group investigation
-discuss in groups
the importance and
necessity of linear
equations and
inequalities and how
it takes place in the
trade.
Practical – solve
linear equations and
simultaneous
equations on a
graph paper
Links to other subjects:Physics (kinematics), Chemistry, Economics...
Assessment criteria: Model and solve algebraically or graphically daily life problems using linear equations or inequalities
Materials: Geometric instruments (ruler-square ....), Digital technology including calculator, ...
Topic Area: ALGEBRA
S4- MATHEMATICS
Sub-topic Area: EQUATIONS AND INEQUALITIES
Unit 6: Quadratic equations and inequalities
No. of lessons:18
Key unit competence: Model and solve algebraically or graphically daily life problems using quadratic equations or inequalities
27
Learning Objectives
Knowledge and
understanding
Skills
Attitudes and values
Content
 Define a quadratic equation
 Be able to solve problems
related to quadratic
equations
 Apply critical thinking
 Appreciate, value
 Equations and

by solving any situation
and care for
inequalities in one
related to quadratic
situations involving
unknown
equations (Economics
to quadratic
 Parametric equations
problems, Finance
equations and
and inequalities in one
problems...)
quadratic
unknown.
 Determine the sign, sum
inequalities in daily  Simultaneous equations
and product of a
life situation
in two unknowns..
quadratic equation
 Show curiosity
 Applications:
 Discuss the parameter
about quadratic
 Physics ( projectile
of a quadratic equation
equations and
motions, …)
and a quadratic
quadratic
 Masonry (Arched
inequality
inequalities
shape,…)
 Explain how to reduce
Bi-quadratic equations
to a quadratic equation
and other equations of
degree greater than 2
Links to other subjects: Physics (Kinematics), Masonry, Economics, Finance...
Assessment criteria: Model and solve algebraically or graphically daily life problems using linear equations or inequalities
Materials: Geometric instruments (Ruler, T-Square ...), Digital technology including calculator
Topic Area: ANALYSIS
S4- MATHEMATICS
Learning Activities
Group investigation:-Discussion
in group the importance and
necessity of a quadratic equation
and a quadratic inequality and
how it take place in Finance
problems Economics problems,
Physics -sketch on graph paper
the graph of quadratic equations
Sub-topic Area: FUNCTIONS
Unit 7: Polynomial, Rational and Irrational functions
No. of lessons:14
Key unit Competence: Use concepts and definitions of functions to determine the domain of rational functions and represent them graphically in
simple cases and solve related problems…
28
Learning Objectives
Knowledge and
understanding





Identify a function as a rule
and recognize rules that are
not functions
Determine the domain and
range of a function
Construct composition of
functions
Find whether a function is
even , odd , or neither
Demonstrate an
understanding of
operations on polynomials,
rational and irrational
functions, and find the
composite of two functions.
Content
Skills
Attitudes and values
 Perform operations on
functions
 Apply different
properties of functions
to model and solve
related problems in
various practical
contexts.
 Analyse, model and
solve problems
involving linear or
quadratic functions and
interpret the results.
 Increase self Factorization of polynomials
confidence and
 Generalities on numerical functions:
determination to
 Definitions
appreciate and
 Domain and range of a function
explain the
 Operations
importance of
 Parity of a function (odd or even)
functions
 Application of rational and irrational
 Show concern on
functions:
patience, mutual
 Physics (Free fall, Projectile…),
respect and
 Economics ( Cost of commodity,
tolerance
or marginal cost, …)
When solving
 Chemistry ( Rate of reaction, …)
problems about
polynomial, rational
and irrational
functions
Learning Activities
 Practical:
Study algebraically
and graphically
polynomial functions.
 Practical: discuss in
groups patiently in
mutual respect and
tolerance, different
operations on
factorizations
 Model or interpret
the problems
related to
polynomial
functions
Links to other subjects: : Physics (eg: Use a quadratic function to model the fall of a ball,…), Economics ( Use of polynomials to represent the cost of producing
“x” units of a commodity, or marginal cost,), Chemistry ( use polynomial to express the rate of reaction in chemistry)
Assessment criteria: Use concepts and definitions of functions to determine the domain of rational functions and represent them graphically in simple cases and
solve related problems…
Materials: Pair of compasses, Graph Papers, ruler, Digital technology ( including calculators,…)
Topic Area: ANALYSIS
S4 - MATHEMATICS
Sub-topic Area: LIMITS, DIFFERENTIATION AND INTEGRATION
Unit 8: Limits of polynomial, rational and irrational functions
29
No. of lessons:14
Key unitCompetence:Evaluate correctly limits of functions and apply them to solve related problems
Learning Objectives
Knowledge and
understanding
 Define the concept of limit
for real-valued functions of
one real variable
 Evaluate the limit of a
function and extend this
concept to determine the
asymptotes of the given
function.
Skills
Attitudes and values
 Calculate limits of
certain elementary
functions
 Develop introductory
calculus reasoning.
 Solve problems
involving continuity.
 Apply informal methods
to explore the concept
of a limit including one
sided limits.
 Use the concepts of
limits to determine the
asymptotes to the
rational and polynomial
functions
 Show concern on
the importance, the
use and
determination of
limit of functions
 Appreciate the use
of
intermediate‐value
theorem
Content
 Concepts of limits:
 Neighborhood of a real number
 Limit of a variable
 Definition and graphical
interpretation of limit of a
function
 One-sided limits
 Squeeze theorem
 Limits of functions at infinity.
 Operations on limits
 Indeterminate cases:
 0
, ,   ,0.
 0
 Applications:
 Continuity of a function at a point
or on interval I
 Asymptotes
Links to other subjects: Physics ( Calculation of velocity, acceleration using concepts of limits)
Assessment criteria: Evaluate correctly limits of functions and apply them to solve related problems
Materials: Manila papers, Graph Papers, ruler, markers, Digital, …
Learning Activities
 Learners discuss in
group to evaluate the
limit of a function at a
point both
algebraically and
graphically, extend
this understanding to
determine the
asymptotes.
 Learners represent
on graph papers
limits of some
chosen functions and
draw the possible
asymptotes
Topic Area: ANALYSIS
Sub-topic Area: LIMITS. DIFFERENTIATION AND INTEGRATION
S4- MATHEMATICS Unit 9: Differentiation of polynomials, rational and irrational functions and their applications No. of lessons: 21
Key unit Competence:Use the gradient of a straight line as a measure of rate of change and apply this to line tangent and normal to curves in
various contexts and use the concepts of differentiation to solve and interpret related rates and optimization problems in various contexts
Learning Objectives
Knowledge and
understanding
Skills
Content
Attitudes and
values
30
Learning Activities
 Evaluate
derivatives of
functions using
the definition of
derivative.
 Define and
evaluate from
first principles
the gradient at a
point.
 Distinguish
between
techniques of
differentiation to
use in an
appropriate
context.
 Use properties of
 Appreciate
 Concepts of derivative of a function:
 Group investigation derivatives to
the use of
determine the gradient of
 Definition
differentiate
gradient as a
different function at a point
 Differentiation from first principles
polynomial,
measure of
using definition of
 High order derivatives
rational ad
rate of change  Rules of differentiation
derivatives, from first
irrational
( economics)
principles, chain rule, and
 Applications of differentiation :
functions
 Appreciate
interpret the results.
 Geometric interpretation of derivatives:
 Use first
the
 Practical task
- Equation of the tangent to a curve
principles to
importance
Represent on graph papers
- Equation of normal to a curve
determine the
and use of
the gradient of a straight
 Mean value theorem for derivatives
gradient of the
differentiatio
line and interpret it
- Lagrange’s theorem
tangent line to a
n in
geometrically in various
 Rolle’s theorem
curve at a point.
Kinematics
practical problems.
 Hospital’s theorem
 Apply the
(velocity,

In group, learner use
 Variations of a function (Maximum and
concepts of and
acceleration )
different techniques of
minimum values, critical points, inflexion points,
techniques of
 Show concern
differentiation to model,
concavity, stationary points,…, Increasing and
differentiation to
on derivatives
analyse and solve rates or
decreasing function)
model, analyse
to help in the
optimization problems.
 Rates of change problems:
and solve rates or
understandin

In group, learners
- Gradient as a measure of rate of change
optimisation
g optimization
determine rate of change
- Kinematic meaning of derivatives
problems in
problems
from practical various
 Optimization problems …
different
problems and interpret the
situations.
results
Links to other subjects: Physics, Economics, ...
Assessment criteria: Use the gradient of a straight line as a measure of rate of change and apply this to line tangent or normal of curves in various contexts and
use the concepts of differentiation to solve and interpret related rates and optimization problems in various contexts
Materials:Manila paper, graph paper, digital technology including calculators,...
Topic Area: Linear Algebra
Sub-topic Area: VECTORS
S4 - MATHEMATICS
Unit 10: Vectors Space of real numbers
No. of lessons:16
Key Unit Competence: Determine the magnitude and angle between two vectors and to be able to plot these vectors and Point out dot product of
two vectors
31
Knowledge and
understanding
 Define the scalar
product of two
vectors
 Give examples of
scalar product
 Determine the
magnitude of a
vector and angle
between two vectors
Learning Objectives
Skills
 Calculate the scalar
product of two
vectors
 Analyse a vector in
term of size.
 Calculate the angle
between two
vectors
Attitudes and
values
 Apply and
transfer the
skills of dot
product,
magnitude to
other area of
knowledge
Content
Learning Activities
 In groups:
 Vector spaces 2 :
 Definitions and Operations on
- Learners discuss about the scalar
vectors.
product of two vectors
2
 Properties of vectors
 Sub-vector spaces
- Determine the magnitude of vector
 Linear combination of vectors
and measure the angle between two
 Basis and Dimension
vectors
 Euclidian Vector space 2
 Dot product and properties
- Learners should be given a task to
 Modulus or Magnitude of
determine the magnitude and angle
vectors
between two vectors and plot these
 Angle between two vectors
vectors and Point out dot product of
two vectors
Links to other subjects: Physics (Dynamics), Geography, ...
Assessment criteria: Determine the magnitude of a vector and the angle between two vectors and to be able to plot these vectors and Point out dot product of
two vectors
Materials: Manila papers, Graph papers, Geometric instruments : rulers , T-square ,Protractors, Computers ...
Topic Area: LINEAR ALGEBRA
S4- MATHEMATICS
Sub-topic Area: LINEAR TRANSFORMATIONS IN 2D
Unit 11:Concepts and Operations on linear transformations in 2D
No. of lessons: 14
Key unit Competence: Determine whether a transformation of IR2 is linear or not ,and Perform operations on linear transformations.
32
Learning Objectives
Knowledge and understanding
Skills
 Define and distinguish
 Perform operations
between linear
on linear
transformations in 2D
transformations in
 Define central symmetry ;
2D
Orthogonal projection of a
 Construct the
vector, Identical
composite of two
transformation
linear
 Define a rotation through an
transformations in
angle about the origin,
2D
reflection in the x axis ,in
 Determine whether
a linear
y axis ,in the line y  x
transformation in
 Show that a linear
2D is isomorphism
transformation is
or not.
isomorphism in 2D or not
 Determine the
analytic expression
of the inverse of an
isomorphism in 2D
Attitudes and values
 Appreciate the
importance and
the use of
operations on
transformation in
2D
 Show curiosity for
the study of
operations on
transformations
in 2D
Content
 Linear
transformation in 2D
 Definitions and
Properties
 Geometric
transformations
 Definitions and
Properties
 Kernel and Range
 Operations on
transformations
Learning Activities
In group:
 Learners will be given a task to
discuss whether each
operation in 2D is linear or not.
 Learners will be given a task,
and be asked to:
- Perform operations on linear
transformations, such as:
- Construct the composite of
two linear transformations in
2D
- Solve f(v)= 0 and determine
kernel and range of f(v)
- Determine whether a linear
transformation in 2D is
isomorphism or not.and
determine the analytic
expression of the inverse of
an isomorphism in 2D
Links to other subjects: Physics ( Quantum physics) ,…
Assessment criteria: Demonstrate that a transformation of IR2 is linear and Perform operations on linear transformations.
Materials: Manila papers, Graph Papers, Geometric instruments( ruler, pair of compasses-square),digital technology including calculators
Topic Area: LINEAR ALGEBRA
S4 - MATHEMATICS
Sub-topic Area: LINEAR TRANSFORMATION IN 2D
Unit 12: Matrices of and determinants of order 2
33
No. of lessons: 12
Key Unit Competence:Use matrices and determinants of order 2 to solve systems of linear equations and to define transformations of 2D
Learning Objectives
Knowledge and
understanding
 Define the order of
a matrix
 Define a linear
transformation in
2D by a matrix
 Define operations
on matrices of
order 2
 Show that a square
matrix of order 2 is
invertible or not
Skills
 Reorganise data into
matrices
 Determine the matrix of a
linear transformation in
2D
 Perform operations on
matrices of order 2
 Construct the matrix of the
composite of two linear
transformations in 2D
 Construct the matrix of the
inverse of an isomorphism
of IR2
 Determine the inverse of a
matrix of order 2
Attitudes and
values
 Appreciate the
importance
and the use of
matrices in
organising
data
Content
Learning Activities
 Matrix of a linear
transformation:
 Definition and Operations
 Matrices of geometric
transformations
 Operations on matrices:
 Equality of matrices
 Show curiosity
 Addition
for the study of
 Multiplication by a scalar
matrices of
 Multiplication of matrices
order 2
 Transpose of a matrix
 Inverse of a square matrix
 Determinant of a matrix of order
2
 Definition
 Applications of determinants
 In group:
- Learners should be given a task
to reorganize given data into
matrices andbe asked to perform
different operations on matrices
by calculating their determinant,
-
Learners in group discuss about
to show how a matrix of order 2
is invertible
 Learners should make research
about the importance and use of
matrices for example in Physics,
Economics, Entrepreneurship,
Sports,...,and report the findings
Links to other subjects:Physics: matrices of Lorenz transformations , Entrepreneurship and in Economics (Organisation of data from sales)
Assessment criteria: Use matrices and determinants of order 2 to solve systems of linear equations and to define transformations
Materials:Manila papers, Markers, …
Topic Area: GEOMETRY
Sub-topic Area: PLANE GEOMETRY
34
S4- MATHEMATICS
Unit 13: Points, straight lines and circles in 2D
No. of lessons:21
Key unit Competence:Determine algebraic representations of lines, straight lines and circles in the 2D
Learning Objectives
Knowledge and
understanding
 Define the
coordinate of
a point in 2-D
Skills
Content
Attitudes and values
 Points in 2D:
 Represent a point and /
 Appreciate that a
or a vector in 2D point is a fixed
 Cartesian coordinates of a point
Calculate the distance
position in a plane
 Distance between two points
between two points in 2D  Show concern
 Mid-points of a line segment
and
the
mid
point
of
a
 Define a
patiently and mutual  Lines in 2D:
segment in 2D
straight line
respect in
 Equations of line:
knowing its:
 Determine equations of a
representations and
- Vector equation
- 2 points
straight line ( Vector
calculations
- Parametric equations
- -Direction
equation,Parametric
 Be accurate in
- Cartesian equation
vector
equation,Cartesian
representations and  Problems on points and straight lines in
- Gradient
equation)
calculations
2D:
 Apply knowledge to find  Manifest a team
 Positions
the centre, radius, and
spirit and think
 Angles
diameter to find out the
critically in problem
 Distance
equation of a circle
solving related to
 Definition of a circle
 Perform operations to
the position of
 Cartesian equation of a circle
determine the
straight lines in 2D
 Problems involving position of a circle
intersection of a circle
and a point or position of circle and lines
and a line
in 2D
Links to other subjects: Physics, Chemistry, Geography, Topography, ...
Assessment criteria: Determine algebraic representations of lines, straight lines and circles in the 2D
Materials :Manila paper, graph paper ,Geometric instruments( ruler, T-square), digital technology including calculators
35
Learning Activities




In group learners discuss
about the distance
between two vectors
Learners represent on
graph some Points,
straight lines and circles
in 2D
Learners will be given a
ask in gorup or
individually, to determine
the magnitude of vector
and measure the angle
between two vectors
Learners represent on
graph papers some
chosen points, lines and /
or circles and determine
their parametric or
Cartesian equations
Topic Area: STATISTICS AND PROBABILITY
S4 - MATHEMATICS
Sub-topic Area: DESCRIPTIVE STATISTICS
Unit 14: Measures of dispersion
No. of lessons: 7
Key Unit Competence: Extend understanding, analysis and interpretation of data arising from problems and questions in daily life to include
the standard deviation.
Learning Objectives
Knowledge and
understanding
 Define the variance,
standard deviation
and the coefficient of
variation
Skills
Content
Attitudes and values
Learning Activities
 Determine the
 Appreciate the
 Variance
In group, learners will be
measures of
importance of
 Standard deviation ( including combined given a task and be asked to:
dispersion of a given
measures of
set of data)
 Discuss about the
statistical series.
dispersion in the  Coefficient of variation
measures of dispersion,
 Apply and explain
interpretation of
 Application:
interpret them and
 Analyse and interpret
the standard
data
represent their findings.
 Problems to include measure of
critically data and
deviation as the
 Show concern on
 Represent data on graph
dispersion and explain the standard
infer conclusion.
more convenient
how to use the
papers, interpret them
deviation as the more convenient
measure of the
standard
and infer conclusion.
measure of the variability in the
variability in the
deviation as
 Make a research on
interpretation of data
interpretation of data
measure of
given problems arising
 Problems to include measure of
 Express the
variability of data.
from various situations
dispersion and express the coefficient
coefficient of
in daily life, investigate
of variation as a measure of the
variation as a
them to include the
spread of a set of data as a proportion
measure of the
standard deviation, and
of its mean.
spread of a set of
represent their findings.
data as a proportion
of its mean.
Links to other subjects:Physics, Biology, Chemistry, Geography, Finance, Economics,…
Assessment criteria: Extend understanding, analysis and interpretation of data arising from problems and questions in daily life to include the standard
deviation.
Materials: Manila papers, Graph Papers, ruler, digital technology including calculators …
36
Topic Area: STATISTICS AND PROBABILITY
S4 - MATHEMATICS
Sub-topic Area: COMBINATORIAL ANALYSIS AND PROBABILITY
Unity 15: Combinatorics
No. of lessons: 18
Key Unit Competence: Use combinations and permutations to determine the number of ways a random experiment occurs
Learning Objectives
Knowledge and
understanding
 Define the
combinatorial
analysis
 Recognize whether
repetition is allowed
or not. and if order
matters on not in
performing a given
experiment
 Construct Pascal’s
triangle
 Distinguish between
permutations and
combinations
Skills
 Determine the
number of
permutations
and
combinations of
“n” items, “r”
taken at a time.
 Use counting
techniques to
solve related
problems.
 Use properties
of combinations
Content
Attitudes and
values
 Appreciate the
importance of
counting
techniques
 Show concern
on how to use
the counting
techniques
 Counting techniques:
 Venn diagram
 Tree diagrams
 Contingency table
 Multiplication principles
 Arrangement and Permutations:
 Arrangements with or without
repetition
 Permutations with or without
repetition
 Combinations:
 Definitions and properties
 Pascal’s triangles
 Binomial expansion
Links to other subjects: English, Physics, Biology, Chemistry, Geography, Finance, Economics, Medical sciences…
Assessment criteria: Calculate accurately combinations or permutations of “n” items , “r” taken at a time.
Materials: Manila papers, Graph Papers, ruler, digital components including calculators …
37
Learning Activities
Learners will be given:
 A mental task, and be asked to
imagine that, if you are a
photographer sitting a group in a
row for pictures. You need to
determine how many different ways
you can seat the group. Learners find
out.
 Questions in group, and be asked
about counting techniques to use for
example “ In how many different
ways could a committee of 5 people
be chosen from a class of 30
students? ”
 A task of using the letters from their
proper words and be asked to create
their own words, E.g:use letters of
“MISSISSIPI”, without a prior
instructions, to create news words
then give feedback..
Topic Area: STATISTICS AND PROBABILITY
S4- MATHEMATICS
Sub-topic Area: COMBINATORIAL ANALYSIS AND PROBABILITY
Unit 16: Elementary Probability
No. of lessons: 7
Key unit Competence:Use counting techniques and concepts of probability to determine the probability of possible outcomes of events
occurring under equally likely assumptions
Learning Objectives
Knowledge and
understanding
 Define probability and
explain probability as a
measure of chance
 Distinguish between
mutually exclusive and
non-exclusive events
and compute their
probabilities
Skills
Attitudes and values
 Use and apply properties
of probability to calculate
the number possible
outcomes of occurring
event under equally likely
assumptions
 Determine and explain
expectations from an
experiment with possible
outcomes
 Appreciate the use of
probability as a measure
of chance
 Show concern on
patience, mutual respect,
tolerance and curiosity in
the determination of the
number of possible
outcomes of a random
experiment
Content
Learning Activities
 Concepts of probability:
 Random experiment
 Sample space
 Event
 Definition of
probability of an
event under equally
likely assumptions
 Properties and formulas
 Learners discuss
gambling problems and
report your result to the
group
 Learners are given a task
of sitting 3 men and 4
women at random in a
row. In groups, they
discuss about the
probability that all the
men are sited together
then they give feedback.
Links to other subjects: Biology (Genetic), Economics, Physics(Quantum Physics), Chemistry,...
Assessment criteria: Use counting techniques and concepts of probability to determine the probability of possible outcomes of occurring event under
equally likely assumptions
Materials: manila papers, Graph Papers, ruler, Digital technology including calculators
38
5.3. MATHEMATICS PROGRAM FOR SECONDARY FIVE
5.3.1. KEY COMPETENCES AT THE END OF SECONDARY FIVE
After completion of secondary 5, the mathematics syallabus will help the learner to:
1. Extend the use of the trigonometric concepts and transformation formulas to solve problems involving
trigonometric equations, inequalities and or trigonometric identities
2. Use arithmetic, geometric and harmonic sequences, including convergence to understand and solve problems
arising in various context.
3. Solve equations involving logarithms or exponentials and apply them to model and solve related problems.
4. Study and to representgraphically a numerical function.
5. Apply theorems of limits and formulas to solve problems involving differentiation including optimization, …
3
3
6. Study linear dependence of vectors of IR and perform operations on linear transformations of IR using vectors.
7. Extend the use of matrices and determinants to order 3 to sove problems in various contexts
8. Use algebraic representations of lines, spheres and planes in 3D space and solve related problems.
9. Extend the understanding, analysis and interpretation of bivariate data to correlation coefficients and regression
lines
10. Solve problems using Bayes theorem and data to make decisions about likelihood and risk.
39
5.3.2. MATHEMATICS UNITS FOR SECONDARY FIVE
Topic Area: TRIGONOMETRY
Sub-topic Area: TRIGONOMETRIC CIRCLE AND IDENTITIES
S5 - MATHEMATICS
Unit 1:Trigonometric functions and equations
No. of lessons:35
Key unit Competence: Solve trigonometric equations, inequalities and related problems using trigonometric functions and equations
Learning Objectives
Contents
Learning Activities
Knowledge and
Skills
Attitudes and values
understanding
 Show how to use
 Apply the transformation
 Appreciate the use and
 Transformation formulas:  In groups, learners
transformation formula
formulas to simply
importance of trigonometric
discuss on how to
 Addition and
to simplify the
trigonometric expressions
functions and equations to
simplify
subtraction formulas
trigonometric
 Use trigonometric functions
understand problems arising in
trigonometric
 Double-angle and halfexpressions
and equations to model and
Complex numbers, in integration
expressions using
angle formulas
 Extend the concepts of
solve problems involving
in Harmonic motion, …
transformation
 Sum, Difference and
trigonometric ratios
trigonometry concepts.
 Appreciate the relationship
formulas – solve
Product Formulas
and their properties to
 Apply trigonometric
between trigonometry and other
problems
 Trigonometric equations
trigonometric
functions and equations to
subjects.
involving
 Trigonometric
equations
perform so far operations in
 Show concern on patience, mutual
trigonometric
inequalities
 Analyze and discuss the
complex numbers,
respect, tolerance and curiosity in
equations and
 Applications:
solution of
calculation in integration and
the solving and discussion about
inequalities
 Euler’s form in
trigonometric
solving problems including
problems involving trigonometric
Complex numbers
inequalities
harmonic motion in Physicsfunctions and equations.
 Integration
Solve any other problems
 Harmonic motion in
related to trigonometry –
Physics
Analyse and explain the
solutions
Links to other subjects: Physics (Harmonic motion), Complex numbers ( Euler’s formula), Analysis ( integration of trigonometric functions )
Assessment criteria: Apply trigonometry functions, transformations formulas and equations to solve problems related to trigonometry.
Materials: Geometric instruments ( ruler,T-square, compass), graph papers, calculators,…
40
Topic Area: ALGEBRA
Sub-topic Area: NUMBER PATTERNS
S5 - MATHEMATICS
Unit 2: Sequences
No. of lessons:25
Key unit Competence: Understand, manipulate and use arithmetic, geometric and harmonic sequences, including convergence.
Learning Objectives
Knowledge and understanding
Skills
Attitudes and values
 Define a sequence and determine if
a given sequence increases or
decreases, converges or not
 Define and understand arithmetic
progressions and their properties
 Determine the value of “n”, given the
sum ofthe first “n”terms of
arithmetic progressions.
 Show how to apply formulas to
determine the “nth” term and the
sum of the first “n”terms of
arithmetic progressions
 Extend the concepts of arithmetic
progression to harmonic sequences
 Define and explaingeometric
progressions and their properties
 Determine the value of “n”, given the
sum ofthe first “n”terms of geometric
progressions
 Show how to apply formulas to
determine specific terms,the “nth”
term and the sum of the first
“n”terms of geometric progressions.
 Use basic concepts and
 Appreciate the
formulas of sequences to
relationship
find the value “n”, given
between the
the sum ofthe first
sequences and
“n”terms of arithmetic
other subjects to
progressions - the “nth”
understand
term and the sum of the
occurring
first “n”terms of
situations (in
arithmetic progressions
Economics: Value
 Use basic concepts and
of annuity, future
formulas of sequences to
value of money; in
find the value “n”, given
Physic: Harmonic
the sum of the first
motion…) values
“n”terms of arithmetic
for different angles
progressions - the “nth”
 Show concern on,
term and the sum of the
patience, mutual
first “n”terms of geometric
respect, tolerance
progressions
and curiosity to
 Apply the concepts of
discuss about
sequences to solve
sequences and
problems involving
their applications.
arithmetic, harmonic or
geometric sequences.
Contents
Learning Activities
 Generalities on
 Group led approach:
sequences
Learners should be given a task
 Arithmetic and
of folding a piece of paper to
harmonic
make them understand the
sequences
meaning of geometric sequences
 Geometric
, and think what should be the
sequences
last term to the infinity
 Applications:
 Problems
including
population
growth
1 1 1
1
 Problems
, , ,... n
2 4 8
2
including
compound and
simple interests  Group investigation :If the
bankrates increase or decrease
 Half-life and
unexpectedly, learners discuss or
Decay problems
investigate how in the next nin Radioactivity
years:
 Bacteria growth
they come out ahead, the deal
problems in
stays fair
Biology …
Links to other subjects: Demography (Populaton growth Problems, Economics (Compound and simple interests), Chemistry (Half-life and Decay problems in
Radioactivity), Biology (Bacteria growth problems) …
Assessment criteria: Apply concepts of sequences to solve problems involving arithmetic, harmonic or geometric sequences
Materials: Geometric instruments (ruler, T-square, compass), graph papers, digital technology including calculators, manila paper,…
41
Topic Area: ALGEBRA
Sub-topic Area: EQUATIONS AND INEQUALITIES
S5 - MATHEMATICS
Unit 3: Logarithmic and exponential equations
No. of lessons: 14
Key unit Competence: Solve equations involving logarithms or exponentials and apply them to model and solve related problems.
Learning Objectives
Knowledge and understanding
Skills
Attitudes and values
 Define logarithm or exponential
equations using properties of
logarithms in any base
 Use the properties of
logarithms to solve
logarithmic and
exponential equations
 State and demonstrate properties of  Convert the logarithm to
logarithms and exponentials
exponential form
 Apply logarithms or
 Carry out operations using the
exponential to solve rates
change of base of logarithms
problems, mortgage
problems, population
growth problems
Contents
Learning Activities
 Appreciate the use
 Logarithmic
In group or individually,
of logarithmic
equations, including
learners:
equations to model
natural logarithms.
and solve problem
 Exponential equations  Once they have the shape of
involving
 Application:
a logarithmic graph, they
logarithms such
can shift it vertically or
 Interest rates
radioactive-decay
horizontally, stretch it,
problems
problems, Carbon
shrink it, reflect it, check
 Mortgage problems
dating problems,
answers with it, and the
 Population growth
problems about
most important is to
problems
alcohol and risk of
interpret the graph.
 Radioactive decay
car accident, etc.

Given for example a growth
problems
 Show concern on
or decay situation, learners
 Earthquake
patience, mutual
after investigating the
problems
respect and
situation, they write an
 Carbon dating
tolerance in solving
exponential function and
problems
problems involving
evaluate it for a given input.
 Problems about
logarithmic or
alcohol and risk of
exponential
car accident
equations
Links to other subjects: Demography ( Population growth Problems), Economics (Interest rates problems, annuity value of money), etc.
Assessment criteria: Apply concepts of logarithmic and exponential equations to solve problems involving logarithms or exponentials.
Materials: Geometric instruments (ruler, T-square, compass), graph papers, digital technology including calculators, Manila paper…
42
Topic Area: ALGEBRA
Sub-topic Area: EQUATIONS AND INEQUALITIES
S5 - MATHEMATICS
Unit 4: Solving equations by numerical method
No. of lessons: 21
Key unit Competence: To be able touse numerical method e.g Newton-Raphson method to approximate solution to equations
Learning Objectives
Knowledge and understanding
Skills
Attitudes and values
 Illustrate numerical
techniques for
approximating solutions to
equations and be aware of
their limitations
 Use numerical methods
to approximate solutions
of equations
 Select a numerical
method appropriate to a
given problem
 Derive error estimates
for approximate
solutions to equations
 Appreciate that
equations can only
be solved
approximately
using numerical
methods
Contents
 Numerical methods:
 Linear Interpolation and
Extrapolation
 Location of Roots:by
graphical and analytical
methods
 Iterative methods:Newton
Raphson Method(General
formula and Tolerance
limit)
 Bisection methods
 Fixed point iteration
Learning Activities
 Work in groups to identify
approximate solutions to
equations that are not easily
solved analytically e.g.
, start by plotting a
graph and getting closer to a
solution via interval bisection.
 In groups research Newton
Raphson method and apply to
the solution of various
equations – identifying when
the method fails
Links to other subjects: Physics ( kinematics), Civil Engineering, Economics, Finance…
Assessment criteria: Use numerical method e.g Newton-Raphson method to approximate solution to equations
Materials: Graph papers, Manila papers, markers, geometric instruments and some software to graph and compute numerical values
43
Topic Area: ANALYSIS
S5 - MATHEMATICS
Sub-topic Area: LIMITS, DIFFERENTIATION AND INTEGRATION
Unit 5: Trigonometric and inverse trigonometric functions
No. of lessons:37
Key unit Competence: Apply theorems of limits and formulas of derivatives to solve problems of refraction of light in prism, Simple harmonic motion problems,
optimization… including trigonometric or inverse trigonometric functions
Learning Objectives
Contents
Learning Activities
Knowledge and
Attitudes and
Skills
understanding
values
 Extend the
 Apply concepts and definition  Appreciate
 Trigonometric functions
 Learners in groups plot the graphs
concepts of
of limits, to calculate the
that questions
of trigonometric functions e.g
 Generalities:
function, domain,
limits of trigonometric
of refraction of
y  sin x or y  a sin bx and
- Definitions
range, period,
functions and remove their
light in prism,
investigate
it, they discuss about
- Domain and range of a
inverse function,
indeterminate forms –
Simple
its period, they find its domain of
function
limits to
Calculate also their high
harmonic
definition and range. – Generalise
- Parity of a function (odd or
trigonometric
derivatives.
motion
this activities to other
even)
functions.
 Derive techniques of
problems,
trigonometric functions. –
- Periodic functions
differentiation to model and
optimization,…
Calculate high derivatives of these
 Limits, including indeterminate
solve problems related to
involving
 Extend the
trigonometric functions.
0
trigonometry.
trigonometric
concepts of limits
cases
, 0.
 Derive techniques of
0
 Apply technique of
functions can
or / and
differentiation to differentiate
differentiation to solve
be solved
differentiation to
 Differentiation of trigonometric
trigonometric or inverse
problems involving
using concepts
modeland solve
functions – Extend this to high
trigonometric functions and apply
trigonometric or inverse
of limits or /
problems
derivatives
them to solve related practical
trigonometric functions such
and
involving

Inverse
trigonometric
functions
problems such as: Refraction of
as refraction of light in prism,
techniques of
trigonometric or
 Applications:
Simple harmonic motion
derivatives.
light in prism;Simple harmonic
inverse
problems,
optimization…

Refraction
of
light
in
prism
trigonometric
motion problems;Etc.
functions.
 Simple harmonic motion
problems
 Etc.
Links to other subjects: Physics (Refraction of light in prism, Simple harmonic motion…) …
Assessment criteria: Apply theorems of limits and formulas of derivatives to solve problems of refraction of light in prism, Simple harmonic motion problems,
optimization… including trigonometric or inverse trigonometric functions
Materials: Geometric instruments (ruler, T-square, compass), graph papers, digital technology including calculators
44
Topic Area: LINEAR ALGEBRA
Sub-topic Area: VECTORS IN 3D
Unit 6: Vector space of real numbers
S5 - MATHEMATICS
Key unit Competence: Study linear dependence of vectors of
vector product to solve mensuration problems in
3
No. of lessons: 21
, solve problems related to angles using the scalar product in
3
and use the
3
Learning Objectives
Knowledge and
Skills
understanding
 Define a basis and the
 Perform operations on
dimension of a vector
vectors in 3 dimensions
space and give examples of  Express a vector of 3 as
bases of 3
linear combination of given
 Define addition of vectors
vectors
3

Determine a basis and the
of
, and multiplication
dimension of a subspace of
of a vector of R3 by a scalar
Contents
Attitudes and values
 Appreciate the
usefulness of
vectors of 3 in the
description of
quantities such as
force, velocity,.. Express a vector of
3
3
 Define the dot product and
as linear
 Calculate the resultant of two
the cross product of two
combination of
vector quantities in threevectors in a threevectors
dimensional space
dimensional vector space
 Appreciate the
 Determine the dot product
and list their properties.
importance of the
and the vector product of
 Define the magnitude of a
dot product as a
two
vectors
in
a
threethree-dimensional vector
measure of
dimensional space and use
and list its properties
parallelism and the
them to solve practical
 Distinguish between the
cross product as a
related problems.
dot product and the cross
measure of
 Explain geometrically the dot
product.
perpendicularity.
product and the cross
product
Links to other subjects: Physics (force, velocity,acceleration), …
45
Learning Activities
 Vector spaces 3 :
 Definitions and
Operations on vectors.




Properties of vectors
Sub-vector spaces
Linear combination of
vectors
Basis and Dimension
3
 Euclidian Vector space 3
 Dot product and
properties
 Modulus or Magnitude of
vectors
 Angle between two
vectors
 Vector product and
properties
Learners perform specific tasks in
group, patiently, in mutual respect
and tolerance such as
 To draw a threedimensional coordinate
system and plot some
chosen points and
represent the
corresponding vectors
 Choose some learners to
simulate points and vectors
in three-dimensional space
and ask the audience to
describe vectors and related
operations
 Study vectors in threedimensional coordinate
system to describe
quantities such as force,
velocity, acceleration,…
Assessment criteria: Apply linear dependence of vectors of 3 to solve problems related to angles using the scalar product in
solve also mensuration problems in 3
Materials: Geometric instruments (ruler, T-square, compass), graph papers, digital technology including calculators
Topic Area: LINEAR ALGEBRA
3
and use the vector product to
Sub-topic Area: LINEAR TRANSFORMATION IN 3D
Unit 7: Matrices and determinants of order 3
S5 - MATHEMATICS
No. of lessons: 35
Key unit Competence: Apply matrix and determinant of order 3 to solve related problems. Demonstrate that a transformation of IR3 is linear and perform
3
operations on linear transformations of IR using vectors.
Learning Objectives
Knowledge and
Skills
understanding
 Define operations on
matrices of order 3
 Illustrate the properties of
determinants of matrices of
order3.
 Show that a square matrix of
order 3 is invertible or not
 Define a linear
transformation in 3D by a
matrix
 Define and perform
operations on linear
transformations of 3
 Express analytically the
inverse of an isomorphism of
3
 Perform operations on matrices of order 3
 Calculate the determinants of matrices of order 3
 Explain using determinant whether a matrix of
order 3 is invertible or not
 Determine the inverse of a matrix of order 3
 Reorganise data into matrices
 Determine the matrix of a linear transformation in
3D
 Apply matrices to solve related problems (e.g in
physics.)
 Apply the concepts of linear transformation to
determine and perform various operations on
linear transformations of 3 .
 Use the properties of linear transformation of
3
to construct the analytic expression of the
inverse of an isomorphism of
3
46
Attitudes and
values
Contents
Learning Activities
 Appreciate
the
importance
of matrices
of order 3
and their
determinant
s in
organising
data and
solving
related
problems.
 Appreciate
the
importance
of
 Matrix of a linear
transformation:
 Definition and
Operations
 Operations on
matrices:
 Equality of matrices
 Addition
 Multiplication by a
scalar
 Multiplication of
matrices
 Transpose of a matrix
 Inverse of a square
matrix
 Determinant of a
matrix of order 3
 Learners discuss in
group patiently, in
mutual respect and
tolerance, how to
organize data into
matrices of order 3
and apply these
concepts to solve
related problems.
 Learners discuss in
group and perform
operations on linear
transformations
of 3 .
 Learners discuss in
group, with respect
to a parameter, the
 Discuss with respect to a
parameter the solutions of a
system of three linear
equations in three unknowns
 Use Cramer’s rule to solve a system of three linear
equations in three unknowns
 Apply properties of determinants to solve
problems related to matrices of order 3.
operations
on linear
transformati
ons of
3
and their
properties.
 Definition
 Applications of
determinants
solutions of a
system of three
linear equations in
three unknowns.
Links to other subjects: Physics (Expressing force, velocity, acceleration,…), Engineering,...
Assessment criteria: Apply matrix and determinant of order 3 to solve related problems. Demonstrate that a transformation of IR3 is linear and perform
3
operations on linear transformations of IR using vectors.
Materials: Geometric instruments (ruler, T-square, compass), graph papers, digital technology including calculators
Topic Area: GEOMETRY
S5 - MATHEMATICS
Sub-topic Area: SPACE GEOMETRY
Unit 8: Points, straight lines and sphere in 3D
Key unit Competence:
Use algebraic representations of points, lines, spheres and planes in 3D space and solve related problems
Learning Objectives
Contents
Knowledge and
Skills
Attitudes and values
understanding
 Define by its
 Represent a point and / or a vector  Appreciate the
 Points in 3D:
coordinates the
in 3D - Calculate the distance
importance of the
 Cartesian coordinates of a point,
position of a
between two points in 3D and the
measure of angle
Distance between two points,
point in 3D
mid - point of a segment in 3D
between two line
Mid-points of a line segment
 Determine equations of a straight
segments.
 Lines in 3D:
line ( Vector equation,Parametric
 Be accurate in
 Equations of line:
 Define a line
equation,Cartesian equation)
calculation to
- Vector, Parametric equations,
using points
 Explain the position vector of a
determine Cartesian
Cartesian equation
and direction
plane - Determine vector equation,
and / or parametric
 Planes in 3D:
vector
parametric equations of a plane
equations of a line –
 Determination of a plane in 3D
and / or the Cartesian equation of
Be aware of the
 Equations of line:
47
No. of lessons: 34
Learning Activities
 Learners represent on
graph papers some points in
3D
 Learners investigate lines,
distance and line segments
and measure angles,
distance between them and
present their findings
a plane - Invent the Cartesian
equation of a plane using
 Define the
parametric equations
position vectors  Apply knowledge to find the
of plane
centre, radius, and diameter to find
out the equation of a sphere
 Perform operations to determine
 Define the
the intersection of a sphere and a
positions of a
plane; intersection of a sphere and
line and a
a line
sphere in 3D
usefulness of the
equations of a plane
 Think critically in
problem solving
related to the
equations of lines and
planes
 Appreciate the
importance of the
distance in space to
determine the
equation of a sphere






Vector, Parametric, Cartesian
equation
Problems on points and straight
lines in 3D:
Positions, Angles, Distance
Sphere
Definition
Cartesian equation of a sphere
Positions of sphere in 3D:
- Point – Sphere
- Line – Sphere
- Plane - Sphere
 Learners represent on
graph papers some chosen
planes and determine its
parametric or Cartesian
equations and present their
findings
 In groups, learners discuss
about the position of
sphere:
- Point – Sphere
- Line – Sphere
- Plane - Sphere
Links to other subjects: Geometry ( distance and positions,…),, Physics ( distances,…), Chemistry ( molecular structures) Engineering ( Architecture), …
Assessment criteria: Solve problems of positions, distances, angles about points, lines and planes, and sphere in 3D.
Materials: Geometric instruments (ruler, T-square, compass), graph papers, digital technology including calculators
Topic Area: STATISTICS AND PROBABILITY
S5 - MATHEMATICS
Sub-topic Area: DESCRIPTIVE STATISTICS
Unit 9: Bivariate statistics
No. of lessons: 10
Key unit Competence: Extend understanding, analysis and interpretation of bivariate data to correlation coefficients and regression lines
Learning Objectives
Contents
Learning Activities
Knowledge and
Skills
Attitudes and values
understanding
 Define the
 Determine the
 Appreciate the
 Covariance
 Learners discuss in groups, about, the correlation
covariance,
coefficient of
importance of
 Correlation
between class results and rank in school for
coefficient of
correlation,
regression lines
coefficient
example. They investigate them, they analysethe
48
correlation and
regression lines.
 Analyse, interpret
data critically then
infer conclusion.
covariance and
regression lines of
bivariate data of
dispersion of a
given statistical
series.
 Apply and explain
the coefficient of
correlation and
standard deviation
as the more
convenient
measure of the
variability in the
interpretation of
data.
and coefficient of
correlation
analyse, interpret
data to infer
conclusion Predict event e.g
after analysing the
population growth
of a given country,
we can make a
decision about the
future generation.
 Regression lines
 Applications:
 Data analysis,
interpretation and
prediction
problems in
various areas
(Biology, Business,
Engineering,
Geography,
Demography …)
relationship between them, and check how the
coefficient of correlation reflects the amount of
variability that is shared between them and what
they have in common. They finally infer conclusion.
 Learners plot visually data on scatter diagram or
scatter plot to represent a correlation between two
variable. – Analyse the graph, infer conclusion using
coefficient of correlation to make predictions about
the variables studied.
 E.g
Links to other subjects:
Geography (spatial statistics research, Air pollution in different year…), Biology (Bio-statistics,..), Chemistry, Demography (Population growth),…
Assessment criteria: Extend understanding, analysis and interpretation of bivariate data to correlation coefficients and regression lines
Materials: Geometric instruments (ruler, T-square, compass), graph papers, digital technology including calculators
49
Topic Area: STATISTICS AND PROBABILITY
S5 - MATHEMATICS
Sub-topic Area: PROBABILITY
Unit 10: Conditional probability and Bayes theorem
No. of lessons: 20
Key unit Competence: Solve problems using Bayes theorem and use data to make decisions about likelihood and risk
Learning Objectives
Contents
Knowledge and
Skills
Attitudes and values
understanding
 Extend the
 Apply theorem of
 Appreciate the use of
 Conditional probability:
concept of
probability to calculate
probability theorem as
 Probability of event B
probability to
the number possible
measure of chance.
occurring when event
explain it as a
outcomes of occurring
 Show concern on
A has already taken
measure of
independent events
patience, mutual respect,
place.
chance.
under equally likely
tolerance and curiosity
 Basic formulae and
 Compute the
assumptions.
about the possible
properties and of
probability of an
 Determine and explain
outcomes of event B
conditional
event B occurring
results from an
occurring when event A
Probability
when event A has
experiment with possible
has already taken place.
 Independent events
already taken
outcomes
 Appreciate the use of
 Probability tree
place.
 Apply Bayes theorem to
Bayes theorem to
diagram
 Interpret data to
calculate the number of
determine the
 Bayes theorem and its
make decision
possible outcomes of
probability of event B
applications
about likelihood
occurring independent
occurring when event A
and risk.
events under equally
has already taken place.
likely assumptions.
Links to other subjects: Geography, Biology, Chemistry, Demography …
Assessment criteria: Solve problems using Bayes theorem and use data to make decisions about likelihood and risk
Materials: Manila papers, ,markers, digital technology including calculators
50
Learning Activities
 Learners discuss in groups
patiently in mutual respect and
tolerance, about number of
possible outcomes of event B
occurring when even A has
already taken place.
 In a given task, learners use
Bayes theorem to determine the
probability of event B occurring
when event A has already taken
place.
5.4. MATHEMATICS PROGRAM FOR SECONDARY SIX
5.4.1. KEY COMPETENCES AT THE END OF SECONDARY SIX
After completion of secondary 6, the mathematics syallabus will help the learnerto:
1. Extend understanding of sets of numbers to complex numbers
2. Solve polynomial equations in the set of complex numbers and solve related problems in physics, …
3. Extend the use of concepts and definitions of functions to determine the domain of logarithmic and exponential
functions.
4. Use integration as the inverse of differentiation and as the limit of a sum and apply them to finding area and volumes
to solve various practical problems.
5. Use differential equations to solve related problems that arise in a variety of practical contexts
6. Relate the sum and the intersection of subspaces of a vector space by the dimension formula
7. Determine the kernel and the image of a linear transformation and use the results to infer the properties of a linear
transformation
8. Transform a matrix to its equivalent form using elementary row operations.
9. Determine algebraic representations of conics in the plane and use them to represent and interpret physical
phenomena.
10. Use probability density functions of a random variable to model events, including binomial, Poisson and Normal
distributions
51
5.4.2. MATHEMATICS UNITS FOR SECONDARY SIX
Topic Area: ALGEBRA
Sub-topic Area: NUMBERS AND OPERATIONS
S6 – MATHEMATICS
Unit 1: COMPLEX NUMBERS
No. of lessons:36
Key unit competence: Perform operations on complex numbers in different forms and use complex numbers to solve related problems in
Physics(voltage and current in alternating current), computer Science(fractals), Trigonometry(Euler’s formula to transform trigonometric
expressions)
Learning Objectives
Knowledge and
understanding
 Identify the real part
and the imaginary
part of a complex
number
 Convert a complex
number from one
form to another
 Represent a complex
number on Argand
diagram
 State De Moivre’s
formula and Euler’s
formula
Skills
 Apply the properties of
complex numbers to
perform operations on
complex numbers in
algebraic form, in polar
form or in exponential
form
 Find the modulus and the
square roots of a complex
number
 Solve in the set of complex
numbers a linear or
quadratic equation
 Use the properties of
complex numbers to
factorize a polynomial and
to solve a polynomial
equation in the set of
complex numbers
 Apply De Moivre’s formula
and Euler’s formula to
Attitudes and
values
 Appreciate the
importance of
complex
numbers in
solving
problems about
alternating
current
 in Physics
,fractals in
Computer
science, etc
 Show
orderliness in
the performance
of tasks about
complex
numbers
52
Content
Learning Activities
 Concepts of complex numbers:
 Definition and structures
 Algebraic structures
 Algebraic form of Complex numbers
 Definition and properties of “ i ”
 Operations:
- Addition, subtraction,
multiplication, powers,
Conjugate and division
 Modulus of a complex number
 Square roots in the set
of complex
numbers
 Equations in the set
of complex
numbers
 Polynomials in the set
of complex
numbers
 Geometric representation of complex
numbers
 Polar form of complex numbers
 Definition
 Modulus and argument of a complex
 Mental work; use definition
of the multiplication of
complex numbers to
determine the complex
number whose square is -1
and draw conclusion about
the properties of “i”
 Constructing the points
representing the nth roots
of a complex number on
Argand diagram and
drawing conclusion about
the polygon obtained by
joining the points
representing the nth roots
 Learners derive
properties of operations
on complex numbers in
trigonometric form and
transform trigonometric
expressions
 Solve linear trigonometric
equations using complex
numbers
number
 Operations
 De Moivre’s formula
 Nth roots of a complex number
 Construction of regular polygons
 Exponential forms of complex numbers:
- Definition and operations
- Euler’s formula of complex
numbers
 Application of complex numbers
 Product to sum formulas in
trigonometry
 Solution of linear trigonometric
equations
 Alternating current problems in
Physics
Links to other subjects: Physics (alternating current), Computer science(fractals) ,...
apply complex numbers to
transform trigonometric
formulas
 Use internet to determine
the generation of fractals
by complex numbers and
print the different shapes
to present in class
Assessment criteria: Perform operations on complex numbers in different forms and use complex numbers to solve related problems in Physics(voltage and
current in alternating current), Computer Science(fractals), trigonometry(Euler’s formula to transform trigonometric expressions),...
Materials: IT equipments...
53
Topic Area: ANALYSIS
Sub-topic Area: LIMITS, DIFFERENTIATION and INTEGRATION
Unit 2: LOGARITHMIC AND EXPONENTIAL
FUNCTIONS
S6 - MATHEMATICS
No. of lessons: 28
Key unit competence: Extend the concepts of functions to investigate fully logarithmic and exponential functions,finding the domain of
definition, the limits,asyrmptotes,variations, graphs , and model problems about interest rates,population growth or decay,magnitude of
earthquake,etc
Learning Objectives
Knowledge and
understanding
Skills
Attitudes and
values
 State the restrictions
on the base and the
variable in a
logarithmic function
 Extendof the
concept of functions
to investigate fully
logarithmic and
exponential
functions
 Perform operations
on logarithmic and
exponential
functions in any
base
 Recall the
differentiation
formulas for
logarithmic and
 Transform a logarithm
from a base to another
 Find the domain and
the range of a
logarithmic or an
exponential function
 Calculate limits of
logarithmic and
exponential functions
 Determine possible
asymptotes of a
logarithmic or an
exponential function
 Determine the
derivative of a
logarithmic or an
exponential function
 Solve related
problems involving
 Show concern on
the importance of
logarithmic and
exponential
functions in
solving problems
such as carbon
dating in
Chemistry
 Develop patience,
dedication and
commitment in
solving problems
about logarithmic
and exponential
functions
 Show mutual
respect,
tolerance, in the
Content
 Logarithmic functions
 Domain of definition
 Limits of logarithmic
functions and their
applications to
continuity and
asymptotes
 Differentiation and its
applications
 Curve sketching
 Exponential functions
 Domain of definition
 Limits of logarithmic
functions and their
applications to
continuity and
asymptotes
 Differentiation and its
applications
54
Learning Activities
 Learners use scientific calculators to
evaluate logarithms and exponentials of
real numbers; they conclude about the
domain (the allowed input values) and
the range(the set of possible outputs)
 Learners may use software,such as
Geogebra, to graph logarithmic and
exponential functions and report to
class their findings about the general
trend of the graphs
 Practical: plot on graph papers, from
the table of values, the points resulting
from logarithmic and exponential
functions and obtain the curves
 Derive formulas about differentiation of
logarithmic and exponential functions
 Discuss in groups the applications of
logarithms and exponentials in real life
and report the results.
 Curve sketching
 Applications of logarithmic
and exponential functions:
 Interest rates problems
 Mortgage problems
 Population growth
problems
 Radioactive decay
problems
 Earthquake problems
 Carbon dating problems
 Problems about alcohol and
risk of car accident.
Links to other subjects: English(as medium of communication), Physics (Newton’s law of cooling), Geography(magnitude of an Earthquake),
Economics(Compounded interest), Biology(population growth), Chemistry(carbon dating) …
exponential
functions to perform
different operations
or solve problems
logarithms
 Sketch the graph of a
logarithmic or an
exponential function
discussion in
group about
logarithmic and
exponential
functions
Assessment criteria: Learner is able to investigate fully logarithmic and exponential functions,finding the domain of definition, the
limits,asyrmptotes,variations, graphs , and model problems about interest rates,population growth or decay,magnitude of earthquake,etc
Materials: Graph Papers, ruler, digital technologies including calculators …
55
Topic Area: ANALYSIS
Sub-topic Area: LIMITS, DIFFERENTIATION AND INTEGRATION
S6 - MATHEMATICS
Unit 3: TAYLOR and MAC-LAURIN’S EXPANSIONS
No. of lessons: 14
Key unit competence: Use Taylor and Maclaurin’s expansion to solve problems about approximations, limits, integration,...
Extend the Maclaurin’s expansion to Taylor series.
Learning Objectives
Knowledge and
understanding
 State the Taylor’s
formula for a
function and the
conditions for the
expansion to be valid
 State the Maclaurin’s
formula for a
function and the
conditions for the
expansion to be valid
 List the first few
terms of Maclaurin’s
expansion of (1+x)n,
cosx, sinx, ex and
.ln(1+x), in each case,
state the conditions
for the expansion to
be valid.
Skills
Attitudes and values
 Derive the Maclaurin’s
series for (1+x)n, cosx,
sinx, ex and .ln(1+x),
 Use Maclaurin’s series to
approximate an irrational
number or the roots of a
transcendental equation
 Apply Maclaurin’s series to
the determination of
asymptotic behavior of a
curve
 Calculate a definite integral
using Maclaurin’s
expansion
 Use properties of
Maclaurin’s expansion to
perform operations on
Maclaurin’s
series(addition,
multiplication by a scalar,
multiplication and division
of functions)
 Show concern on the
importance of Taylor
and Maclaurin’s and
Taylor’s series in
solving problems
about approximation,
limits,...
 Appreciate the
contribution of each
member in the
discussion about
Taylor and
Maclaurin’s
series,show mutual
respect and tolerance
in the discussion in
group
Content
 Generalities on series
 Definition
 Convergence of
series
 Power series:
 Definition and
properties
 Taylor series
 Maclaurin series
 Applications:
 Approximation of an
irrational number
by a rational
number
 Approximation of
the roots of
equations
 Calculation of limits
 Evaluating definite
integrals, etc
Learning Activities
In group:
 Use graphical approach to
demonstrate the behaviour of
different series, to discover Taylor
polynomials and to investigate the
relationship between the functions
they approximate
 Derive different Taylor and
Maclaurin’s formulas and use them
to approximate the value of an
irrational number, solutions of
transcendental equations,
asymptotic behaviour of a curve,
approximation of definite integral,
etc. and report the result to class
Links to other subjects: English(terminologies such as series, convergence), Physics ( path of a moving particle …)
Assessment criteria: Learner is able to Use Taylor and Maclaurin’s expansion to solve problems about approximations, limits, integration
Materials: Graph paper, scientific calculators
56
Topic Area: ANALYSIS
Sub-topic Area: LIMITS, DIFFERENTIATION and INTEGRATION
S6 - MATHEMATICS
Unit 4: INTEGRATION
No. of lessons: 42
Key unit competence: Use integration as the inverse of differentiation and as the limit of a sum then apply it to findarea of plane surfaces, volumes
of solid of revolution, lengths of curved lines.
Learning Objectives
Knowledge and
understanding
 Define the
differential of a
function
 Interpret
geometrically the
differential of a
function
 List the
differentiation
formulas
 Clarify the
relationship
between
derivative and
antiderivative of a
function
 Illustrate the use
of basic
integration
formulas
 Extend the
concepts of
indefinite
integrals to
definite integrals.
Skills
 Use differentials to
approximate a function
and to calculate the
percentage error in an
estimation
 Calculate integrals.
Using appropriate
techniques
 Use properties of
integrals to simplify
the calculation of
integrals
 Calculate a limit of a
sum to infinity as a
definite integral
 Apply definite integrals
to calculate the area,
volume, arc length
 Analyze the
convergence of an
improper integral
 Use integrals to solve
problems in Physics
(work,..), Economics
(marginal and total
cost), etc.
Attitudes and
values
 Show
concern on
the
importance
of integral
calculus in
solving
problems
from daily
life.
 Appreciate
various
techniques of
integration
and show
patience,
commitment
and
tolerance in
the
evaluation of
integrals
Content
 Differentials
 Definitions and Operations on increments.
 Properties of differentials.
 Applications:
- Approximation
- Calculation of error
 Indefinite integrals
 Antiderivatives
 Definition and properties
 Techniques of integration:
- Basic Integration Formulas
- Integration by change of variables
- Integration by Parts
- Integration of Rational Functions by
Partial Fractions
- Integration of trigonometric functions
 Definite integrals
 Definition
 Properties
 Techniques of integration
 Applications of definite integrals
- Calculation of area of a plane surface
- Calculation of volume of a solid of
revolution
- Calculation of length of curved lines
- Applications in Physics (Work,
57
Learning Activities
 Graphical approach:
Considering consecutive
subintervals, learners
calculate the areas of
corresponding rectangles
,then introduce the
concept of integral as sum
to infinity, when the width
tends to zero
 Learners determine the
area of square, rectangle,
trapezium, triangle and
circle – determine also the
volume of sphere, cylinder
and cone – Calculate the
circumference of circle
Energy,…)
 Improper integrals:
 Definition
 Calculation of improper integrals
Links to other subjects: English, Physics (work, Energy) …
Assessment criteria: Use integration as the inverse of differentiation and as the limit of a sum then apply it to findarea of plane surfaces, volumes of solid of
revolution, lengths of curved lines.
Materials: Manila papers, calculators, graph papers, ruler, markers …
58
Topic Area: ANALYSIS
Sub-topic Area: LIMITS, DIFFERENTIATION and INTEGRATION
S6 - MATHEMATICS
Unit 5: DIFFERENTIAL EQUATIONS
No. of lessons: 21
Key unit competence: Use ordinary differential equations of first and second order to model and solve related problems tin Physics, Economics,
Chemistry, Biology,
Learning Objectives
Knowledge and
understanding
 Extend the concepts of
differentiation and
integration to ordinary
differential equations
 State the order and the
degree of an ordinary
differential equation
 Express the auxiliary
quadratic equation of a
homogeneous linear
differential equation of
second order with
constant coefficients
 Predict the form of the
particular solution of
an ordinary linear
differential equation of
second order
Skills
Attitudes and values
 Determine whether an
ordinary differential
equation of first order is
with separable
variables, homogeneous
or linear
 Use appropriate method
to solve an ordinary
differential equation of
first order
 Solve an ordinary linear
differential equation of
first order by “variation
of constant” and by
“integrating factor”
 Solve an ordinary linear
differential equation of
second order
 Use differential
equations to model and
solve problems in
Physics (simple
harmonic motion,…),
Economics (point
elasticity,…),etc.
 Appreciate the use
of differential
equations in solving
problems occurring
from daily life
 Show patience,
commitment and
dedication when
solving a
differential
equation or
modelling a
problem using
differential
equations
 When discussing in
groups the solution
of a differential
equation, make
sense of other
learners’ thinking,
show tolerance and
mutual respect.
Content
 Definition and classification
 1st Order differential
equation
 Differential equation with
separable variables
 Simple homogeneous
differential equations
 Linear differential
equations
 Applications
 2nd Order differential
equation
 Linear equations with
constant coefficients:
- The right hand side
is equal to zero
- The right hand side
is a polynomial
function
- The right hand side
is a trigonometric
function
- The right hand side
is an exponential
function
Learning Activities
 Mental task: imagine the motion of
a child on a swing. Express the
displacement as function of time.
Differentiate the function to find
the velocity and acceleration, and
then express the relation between
the function and its derivatives.
Report your results.
 Use graph plotting to illustrate the
general solution of a differential
equation
 Discuss in groups the solutions of a
differential equation with respect
to a parameter and present the
result to the class, show ability to
communicate your thinking and
reasoning
 Use internet to find the
applications of differential
equations in sciences and report
your findings to the class
 Derive the general solutions of
differential equations
 Solve ordinary differential
equations of first and second
orders
Links to other subjects: Physics (simple harmonic motion), English (integrating factor), Chemistry (radioactive decay), Economics(point elasticity and demand
function),etc…
59
Assessment criteria: Learner is able to Use ordinary differential equations of first and second order to model and solve related problems tin Physics,
Economics, Chemistry, Biology, etc.
Materials: Geometric instruments, graph papers, calculators, ICT equipments
Topic Area: LINEAR ALGEBRA
Sub-topic Area: VECTORS
S6- MATHEMATICS
Unit 6: INTERSECTION AND SUM OF SUBSPACES
No. of lessons:14
Key unit competence: Relate the sum and the intersection of subspaces of a vector space by the dimension formula
Learning Objectives
Knowledge and
understanding
 Define the
intersection
and the sum of
subspaces of a
vector space
 State the
dimension
formula
 List the
conditions for a
vector space to
be qualified as
direct sum of
its subspaces
Skills
Attitudes and values
 Perform the intersection and the
addition of subspaces of a vector
space
 Determine a basis of the
intersection and a basis of the sum
of the subspaces of a vector space
 Apply the concepts of bases to
calculate the dimension of the
intersection and the dimension of
the sum of given subspaces of a
vector space
 Apply the concept of the dimension
formula to determine whether a
vector space is direct sum or not of
its given subspaces
 Appreciate the
relationship
between the
dimensions of the
intersection, the sum
and the subspaces of
a vector space
 Show patience,
commitment and
dedication when
verifying through
the dimension
formula whether a
vector space is direct
sum of its subspaces
or not
Content
Learning Activities
 Intersection of
subspaces
 Definition
 Dimension of
the intersection of
subspaces
 Sum of subspaces
 Definition
 Dimension of
the sum of
subspaces
 Dimension formula
 Direct sum of
subspaces
 Draw a Venn diagram for the vector
space and its subspaces, and shade in
color the intersection of the
subspaces
 Study in group the properties of the
intersection and the sum of subspaces
and present to the class the result
with the ability to communicate the
thinking and reasoning
 Demonstrate that the intersection
and the sum of two subspaces of a
vector space is a subspace of the
vector space
 Construct two subspaces of a given
vector space and show that their
union is not a subspace of the given
vector space
Links to other subjects: Geometry(vectors), Physics(vectors),Engineering(vectors) …
Assessment criteria: Relate the sum and the intersection of subspaces of a vector space by the dimension formula
Materials: Manila papers, markers …
60
Topic Area: LINEAR ALGEBRA
Sub-topic Area: LINEAR TRANSFORMATIONS
S6 - MATHEMATICS
Unit 7: TRANSFORMATION OF MATRICES
No. of lessons: 29
Key unit competence: Transform matrices to an echelon form or to diagonal matrix and use the results to solve simultaneous linear equations or
to calculate the nth power of a matrix
Learning Objectives
Knowledge and
understanding
 Define the kernel,
the image, the
nullity and the rank
of a linear
transformation
 State the dimension
formula for linear
transformations
 Carry out the
elementary row
operations on
matrices
 Define Eigen values
and eigenvectors of
a square matrix
 Discuss the
diagonalisation of a
matrix of order 2
Skills
 Use definitions and
properties to determine
the kernel and the image
of linear transformation
 Apply the concept of
dimension to calculate the
nullity and the rank of
linear transformation
 Use elementary row
operations to obtain a
matrix in echelon form
and to solve simultaneous
linear equations
 Determine a basis in
which the matrix of a
linear transformation of
R2 is diagonal and use the
transition matrix to verify
the result
Attitudes and
values
 Appreciate the
use of the
transformation
of matrices in
the
simplification of
the expression of
a linear
transformation
 Verify the
correctness of
the diagonal
matrix to
develop
selfesteem
Content
Kernel and range
 Definitions
 Nullity and rank of a linear
mapping
 Dimension formula for linear
mapping
 Elementary row operations
 Elementary row operations
 Row reducing matrices to echelon
form and applications
- inverse of a matrix
- Gauss elimination
 Diagonalisation of matrices of order 2
 Characteristic polynomial
 Eigen values ,Eigen spaces and
Eigen vectors of a matrix of order
2
 Diagonalisation of matrices of
order 2
 Power of a matrix

Learning Activities
In group:
 Learners discuss in pair, the
transformation of matrices
to diagonal matrices and
they report, represent the
results
 Derive the formulas and
determine the diagonal
matrices
 Calculate the nth power of a
square matrix after
transforming the matrix to
diagonal form
 Make research using
internet, library,…, about
transformation of matrices
and their applications in
real life and report, present
results to class.
Links to other subjects: English(terminologies ), Geometry (matrices of plane transformations),Economics(organisation of data).
Assessment criteria: Transform matrices to an echelon form or to diagonal matrix and use the results to solve simultaneous linear equations or to calculate the
nth power of a matrix
Materials: Manila papers, Calculators, …
61
Topic Area: GEOMETRY
Sub-topic Area: PLANE GEOMETRY
S6 - MATHEMATICS
Unit 8: CONICS
No. of lessons:35
Key unit competence:
 Determine the characteristics and the graph of a conic given by its Cartesian, parametric or polar equation
 Find the Cartesian, parametric and polar equations of a conic from its characteristics
Learning Objectives
Knowledge and
understanding
 Define geometrically
a conic as the
intersection of a
plane and a cone
and classify conics
from the position of
the intersecting
plane
 Express , in
Cartesian form, the
standard equation of
a parabola, an
ellipse and a
hyperbola
 Classify conics
basing on the value
of the eccentricity
Skills
Attitudes and values
 Find the
characteristics of a
conic given its
equation (centre, axes,
vertices, tangent,
normal, shapes,
asymptotes, foci,
directrices,
eccentricity , etc.)
 Determine the
Cartesian, parametric
and polar equations of
a conic from its
characteristics
 Derive the standard
equation of a conic, in
standard form.
 Use diagonalisation of
matrices to simplify
the equation of an
ellipse or a hyperbola.
 Show patience,
determination when
analysing the general
equation of a conic
 Appreciate the
importance of conics in
the naturally occurring
shapes, such rainbow,
the modelling of paths of
particles such as
projectiles, satellites,
and manmade shapes
such aerial, bridge, etc.
62
Content
Learning Activities
 Parabola :
 Definition
 Cartesian and parametric
equations
 Graphical representation of a
parabola
 Characteristics of a parabola
(Vertex, axis of symmetry, focus,
directrix, tangent and normal, ...)
 General equation
 Ellipse:
 Definition
 Cartesian and parametric
equations
 Graphical representation of an
ellipse
 Characteristics of an ellipse
(Centre of symmetry, Vertices,
axes of symmetry, Eccentricity,
foci, directrices, tangent and
normal, ...)
 General equation
 Visual approach:
Learners analyze conics
from sections of planes
and cones
 Matching conics to their
standard forms
 Analyzing and graphing
conics from their
equations
 Deriving formulas
related to conics( e.g.
equation of the tangent)
 Analyse the general
equation of an ellipse
 Hyperbola:
 Definition
 Cartesian and parametric
equation
 Graphical representation of a
hyperbola
 Characteristics of a hyperbola
(Centre of symmetry, Vertices,
axes of symmetry, asymptotes,
Eccentricity, foci, directrices,
tangent and normal, ...)
 General equation
 Polar coordinates:
 Definition
 Conversion of polar coordinates
to Cartesian coordinates and
vice versa
 Straight line, circle and conics in
polar coordinates
Links to other subjects:Architecture (shape of bridge), Physics(path of particle), Chemistry, Astronomy(motion of satellites),...
Assessment criteria: Determine the characteristics and the graph of a conic given by its Cartesian, parametric or polar equation
Find the Cartesian, parametric and polar equations of a conic from its characteristics
Materials:Geometric instruments (ruler, compass, T-square), IT equipments, conic sections from cones and planes,...
63
and a hyperbola to
obtain the graph
Topic Area: STATISTICS AND PROBABILITY
S6 - MATHEMATICS
Sub-topic Area: PROBABILITY
Unit 9: RANDOM VARIABLES
No. of lessons: 33
Key unit Competence: Calculate and interpret the parameters of a random variable (discrete or continuous) including binomial and the Poisson
distributions
Learning Objectives
Content
Learning Activities
Knowledge and
Attitudes and
Skills
understanding
values
 Define a random
 Use the concepts of statistics
 Appreciate
 Discrete and finite random
 Mental task:
variable
to compare frequency
the use of the
variable:
Imagine a policeman recording
 Identify whether a
distribution to probability
random
the numbers of cars crossing a
 Probability distribution
given random
distribution
variable in
certain junction in different
 Expected value, variance and
variable is discrete or  Calculate and interpret the
the
periods of the day, the nurse
standard deviation of a
continuous
parameters of a random
interpretatio
recording the weights of
discrete random variable
 Define the
variable(discrete or
n of
infants at birth in a hospital
 Cumulative distribution
parameters of a
continuous)
statistical
Classify the random variables
function
discrete random
 Construct the probability
data
as continuous or discrete
 Binomial and
variable
distribution of a discrete

Construction of probability
 The Poisson distribution
 Learn in which
random variable
distribution of discrete random
situation the
 Determine whether a function
variable
 Continuous random variables
Binomial distribution
can serve as probability

Calculation of the expected
 Probability density function
applies and state its
density function or not
value, the variance and the
 Expected value, variance and
parameters, …
 Apply Binomial and The
standard deviation
standard deviation of a
Poisson distributions to solve
 Graphing the probability
discrete random variable
related problems
distribution
Links to other subjects: Economics, Entrepreneurship, Biology, geography (analysis of data)
Assessment criteria: Calculate and interpret the parameters of a random variable(discrete or continuous) including binomial and the Poisson distributions
Materials: Calculator, graph paper, Manila Paper, …
64
6. REFERENCES
1. Alpers, B. (2013). A Framework for Mathematics Curricula in Engineering Education: A Report of the Mathematics
Working Group. European Society for Engineering Education (SEFI): Brussels
2. Balakrishnan, V.K . (1995). Schaum’s Outline, Combinatorics.
3. Bird, J. ( 2003). Engineering Mathematics ( 4th Edition).
4. Bronson, R. (2003). Schaum’s Outline, Differential equation.
5. CrashwshaW,J.,& Chambers,J.(2001).Advanced Level Statistics with worked examples (Fourth Edition). Nelson Thornes
Ltd:UK
6. Curriculum Planning and Development Division (2006). Secondary Mathematics Syllabuses. Ministry of Education:
Singapore
7. Frank Ayres, F., & Moyer,E.R.(1999). Schaum’ s Outline Series Trigonometry (3rd Edition).
8. National Curriculum Development Centre (2008).Mathematics teaching syllabus: Uganda Certificate of Education
Senior1-4. Ministry of Education and Sports: Uganda.
9. National Curriculum Development Centre (2013). Uganda Advanced Certificate of Education: Teaching Syllabi for
Physics and Mathematics (Volume 2). Ministry of Education and Sports: Uganda.
10. National Curriculum Development Centre (2013). Uganda Advanced Certificate of Education: Subsidiary Mathematics
teaching syllabus. Ministry of Education and Sports: Uganda.
11. Rich, B. & Thomas, C. (2009). Schaum’s Outline, Geometry : Fourth Edition.
12. Rwanda Education Board (2010).Advanced Level Mathematics for Science Combination. Ministry of Education: Rwanda
13. Rwanda Education Board (2014). Mathematics Curriculum For Physics - Chemistry - Biology (PCB) Combination
(Elective) Advanced Level. Ministry of Education: Rwanda
14. Sanchez , A.V, & Ruiz, M.P. (2008). Competence-Based Learning : A proposal for the assessment of Generic
Competences(Eds).
15. Shampiyona, A. (2005). Mathématiques 6. Imprimerie de Kigali: Kigali
16. Spiegel,M. R.; Schiller, J. & Srinivasan, A. (2009). Schaum’s Outline, Probability and Statistics: Third Edition.
17. Stewart, J. ( 2008 ). Calculus ( 6th Edition)
18. Stewart, J. (2008). Calculus Early Transcendental, sixth edition. McMaster University: US.
19. Wrede,R., Murray,D., & Spiegel, R. (2010). Schaum’s Outline, Advanced calculus, 3rd Edition.
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7. APPENDIX: SUBJECTS AND WEEKLY TIME ALOCATION FOR A’LEVEL
Subjects
Mathematics
Physics
Computer Science
Chemistry
Biology
Geography
History
Economics
Literature in English
Kinyarwanda major
Kiswahili major
Religion major
Entrepreneurship
General
Studies
Communication
Subsidiary Mathematics
Minor
English
Subjects
French
Kinyarwanda
Kiswahili
and
S4
7
7
7
7
7
7
7
7
7
7
7
7
6
Weekly periods
(1 period = 40 min.)
S5
7
7
7
7
7
7
7
7
7
7
7
7
6
S6
7
7
7
7
7
7
7
7
7
7
7
7
6
3
3
3
3
4
4
4
4
3
4
4
4
4
3
4
4
4
4
66